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Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic geometry and continued fractiontheory I

Ian Short 9 February 2010

httpmathsorg ims25mathspresentationsphp

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Collaborators

Meira Hockman

Alan Beardon(University of Cambridge)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Collaborators

Meira Hockman

Alan Beardon(University of Cambridge)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Collaborators

Meira Hockman

Alan Beardon(University of Cambridge)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Project

The geometry of continued fractions

httpmathsorg ims25mathspresentationsphp

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Project

The geometry of continued fractions

httpmathsorg ims25mathspresentationsphp

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Continued fractions

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Continued fractions

K(an| bn) =a1

b1 +a2

b2 +a3

b3 +a4

b4 + middot middot middot

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Continued fraction convergence

a1

b1

a1

b1 +a2

b2

a1

b1 +a2

b2 +a3

b3

a1

b1 +a2

b2 +a3

b3 +a4

b4

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Continued fraction convergence

a1

b1

a1

b1 +a2

b2

a1

b1 +a2

b2 +a3

b3

a1

b1 +a2

b2 +a3

b3 +a4

b4

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Continued fraction convergence

a1

b1

a1

b1 +a2

b2

a1

b1 +a2

b2 +a3

b3

a1

b1 +a2

b2 +a3

b3 +a4

b4

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Continued fraction convergence

a1

b1

a1

b1 +a2

b2

a1

b1 +a2

b2 +a3

b3

a1

b1 +a2

b2 +a3

b3 +a4

b4

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Expansion of e = 271828182845905 (Euler)

e = 2 +1

1 +1

2 +1

1 +1

1 +1

4 +1

1 +1

1 +1

6 +1

1 +1

1 +1

8 + middot middot middot

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Expansion of π = 314159265358979 (Lange)

π = 3 +12

6 +32

6 +52

6 +72

6 +92

6 + middot middot middot

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

A problematic example

1

2 +1

3 +1

1 +1

minus1 + middot middot middot

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius transformations

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

tn(infin) = 0

Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius transformations

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

tn(infin) = 0

Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius transformations

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

tn(infin) = 0

Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius transformations

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

tn(infin) = 0

Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence again

Convergence of K(an| bn) equivalent to convergence ofT1(0) T2(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Dealing with infin

Previously

1infin

= 0

Now

h(z) =1z

h(infin) = 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Dealing with infin

Previously

1infin

= 0

Now

h(z) =1z

h(infin) = 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps Allows us to bring in geometry Simpler notation composition of maps rather than

algebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps

Allows us to bring in geometry Simpler notation composition of maps rather than

algebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps Allows us to bring in geometry

Simpler notation composition of maps rather thanalgebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps Allows us to bring in geometry Simpler notation composition of maps rather than

algebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Reminder

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Reminder

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions1swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions2swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries

Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group (and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Isometry group

Isom(H3) =M

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Back to continued fractions

tn(infin) = 0 Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Collaborators

Meira Hockman

Alan Beardon(University of Cambridge)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Collaborators

Meira Hockman

Alan Beardon(University of Cambridge)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Collaborators

Meira Hockman

Alan Beardon(University of Cambridge)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Project

The geometry of continued fractions

httpmathsorg ims25mathspresentationsphp

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Project

The geometry of continued fractions

httpmathsorg ims25mathspresentationsphp

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Continued fractions

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Continued fractions

K(an| bn) =a1

b1 +a2

b2 +a3

b3 +a4

b4 + middot middot middot

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Continued fraction convergence

a1

b1

a1

b1 +a2

b2

a1

b1 +a2

b2 +a3

b3

a1

b1 +a2

b2 +a3

b3 +a4

b4

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Continued fraction convergence

a1

b1

a1

b1 +a2

b2

a1

b1 +a2

b2 +a3

b3

a1

b1 +a2

b2 +a3

b3 +a4

b4

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Continued fraction convergence

a1

b1

a1

b1 +a2

b2

a1

b1 +a2

b2 +a3

b3

a1

b1 +a2

b2 +a3

b3 +a4

b4

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Continued fraction convergence

a1

b1

a1

b1 +a2

b2

a1

b1 +a2

b2 +a3

b3

a1

b1 +a2

b2 +a3

b3 +a4

b4

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Expansion of e = 271828182845905 (Euler)

e = 2 +1

1 +1

2 +1

1 +1

1 +1

4 +1

1 +1

1 +1

6 +1

1 +1

1 +1

8 + middot middot middot

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Expansion of π = 314159265358979 (Lange)

π = 3 +12

6 +32

6 +52

6 +72

6 +92

6 + middot middot middot

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

A problematic example

1

2 +1

3 +1

1 +1

minus1 + middot middot middot

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius transformations

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

tn(infin) = 0

Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius transformations

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

tn(infin) = 0

Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius transformations

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

tn(infin) = 0

Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius transformations

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

tn(infin) = 0

Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence again

Convergence of K(an| bn) equivalent to convergence ofT1(0) T2(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Dealing with infin

Previously

1infin

= 0

Now

h(z) =1z

h(infin) = 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Dealing with infin

Previously

1infin

= 0

Now

h(z) =1z

h(infin) = 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps Allows us to bring in geometry Simpler notation composition of maps rather than

algebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps

Allows us to bring in geometry Simpler notation composition of maps rather than

algebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps Allows us to bring in geometry

Simpler notation composition of maps rather thanalgebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps Allows us to bring in geometry Simpler notation composition of maps rather than

algebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Reminder

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Reminder

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions1swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions2swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries

Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group (and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Isometry group

Isom(H3) =M

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Back to continued fractions

tn(infin) = 0 Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Collaborators

Meira Hockman

Alan Beardon(University of Cambridge)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Collaborators

Meira Hockman

Alan Beardon(University of Cambridge)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Project

The geometry of continued fractions

httpmathsorg ims25mathspresentationsphp

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Project

The geometry of continued fractions

httpmathsorg ims25mathspresentationsphp

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Continued fractions

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Continued fractions

K(an| bn) =a1

b1 +a2

b2 +a3

b3 +a4

b4 + middot middot middot

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Continued fraction convergence

a1

b1

a1

b1 +a2

b2

a1

b1 +a2

b2 +a3

b3

a1

b1 +a2

b2 +a3

b3 +a4

b4

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Continued fraction convergence

a1

b1

a1

b1 +a2

b2

a1

b1 +a2

b2 +a3

b3

a1

b1 +a2

b2 +a3

b3 +a4

b4

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Continued fraction convergence

a1

b1

a1

b1 +a2

b2

a1

b1 +a2

b2 +a3

b3

a1

b1 +a2

b2 +a3

b3 +a4

b4

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Continued fraction convergence

a1

b1

a1

b1 +a2

b2

a1

b1 +a2

b2 +a3

b3

a1

b1 +a2

b2 +a3

b3 +a4

b4

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Expansion of e = 271828182845905 (Euler)

e = 2 +1

1 +1

2 +1

1 +1

1 +1

4 +1

1 +1

1 +1

6 +1

1 +1

1 +1

8 + middot middot middot

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Expansion of π = 314159265358979 (Lange)

π = 3 +12

6 +32

6 +52

6 +72

6 +92

6 + middot middot middot

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

A problematic example

1

2 +1

3 +1

1 +1

minus1 + middot middot middot

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius transformations

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

tn(infin) = 0

Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius transformations

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

tn(infin) = 0

Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius transformations

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

tn(infin) = 0

Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius transformations

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

tn(infin) = 0

Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence again

Convergence of K(an| bn) equivalent to convergence ofT1(0) T2(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Dealing with infin

Previously

1infin

= 0

Now

h(z) =1z

h(infin) = 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Dealing with infin

Previously

1infin

= 0

Now

h(z) =1z

h(infin) = 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps Allows us to bring in geometry Simpler notation composition of maps rather than

algebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps

Allows us to bring in geometry Simpler notation composition of maps rather than

algebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps Allows us to bring in geometry

Simpler notation composition of maps rather thanalgebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps Allows us to bring in geometry Simpler notation composition of maps rather than

algebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Reminder

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Reminder

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions1swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions2swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries

Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group (and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Isometry group

Isom(H3) =M

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Back to continued fractions

tn(infin) = 0 Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Collaborators

Meira Hockman

Alan Beardon(University of Cambridge)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Project

The geometry of continued fractions

httpmathsorg ims25mathspresentationsphp

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Project

The geometry of continued fractions

httpmathsorg ims25mathspresentationsphp

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Continued fractions

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Continued fractions

K(an| bn) =a1

b1 +a2

b2 +a3

b3 +a4

b4 + middot middot middot

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Continued fraction convergence

a1

b1

a1

b1 +a2

b2

a1

b1 +a2

b2 +a3

b3

a1

b1 +a2

b2 +a3

b3 +a4

b4

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Continued fraction convergence

a1

b1

a1

b1 +a2

b2

a1

b1 +a2

b2 +a3

b3

a1

b1 +a2

b2 +a3

b3 +a4

b4

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Continued fraction convergence

a1

b1

a1

b1 +a2

b2

a1

b1 +a2

b2 +a3

b3

a1

b1 +a2

b2 +a3

b3 +a4

b4

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Continued fraction convergence

a1

b1

a1

b1 +a2

b2

a1

b1 +a2

b2 +a3

b3

a1

b1 +a2

b2 +a3

b3 +a4

b4

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Expansion of e = 271828182845905 (Euler)

e = 2 +1

1 +1

2 +1

1 +1

1 +1

4 +1

1 +1

1 +1

6 +1

1 +1

1 +1

8 + middot middot middot

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Expansion of π = 314159265358979 (Lange)

π = 3 +12

6 +32

6 +52

6 +72

6 +92

6 + middot middot middot

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

A problematic example

1

2 +1

3 +1

1 +1

minus1 + middot middot middot

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius transformations

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

tn(infin) = 0

Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius transformations

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

tn(infin) = 0

Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius transformations

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

tn(infin) = 0

Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius transformations

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

tn(infin) = 0

Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence again

Convergence of K(an| bn) equivalent to convergence ofT1(0) T2(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Dealing with infin

Previously

1infin

= 0

Now

h(z) =1z

h(infin) = 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Dealing with infin

Previously

1infin

= 0

Now

h(z) =1z

h(infin) = 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps Allows us to bring in geometry Simpler notation composition of maps rather than

algebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps

Allows us to bring in geometry Simpler notation composition of maps rather than

algebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps Allows us to bring in geometry

Simpler notation composition of maps rather thanalgebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps Allows us to bring in geometry Simpler notation composition of maps rather than

algebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Reminder

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Reminder

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions1swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions2swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries

Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group (and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Isometry group

Isom(H3) =M

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Back to continued fractions

tn(infin) = 0 Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Project

The geometry of continued fractions

httpmathsorg ims25mathspresentationsphp

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Project

The geometry of continued fractions

httpmathsorg ims25mathspresentationsphp

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Continued fractions

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Continued fractions

K(an| bn) =a1

b1 +a2

b2 +a3

b3 +a4

b4 + middot middot middot

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Continued fraction convergence

a1

b1

a1

b1 +a2

b2

a1

b1 +a2

b2 +a3

b3

a1

b1 +a2

b2 +a3

b3 +a4

b4

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Continued fraction convergence

a1

b1

a1

b1 +a2

b2

a1

b1 +a2

b2 +a3

b3

a1

b1 +a2

b2 +a3

b3 +a4

b4

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Continued fraction convergence

a1

b1

a1

b1 +a2

b2

a1

b1 +a2

b2 +a3

b3

a1

b1 +a2

b2 +a3

b3 +a4

b4

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Continued fraction convergence

a1

b1

a1

b1 +a2

b2

a1

b1 +a2

b2 +a3

b3

a1

b1 +a2

b2 +a3

b3 +a4

b4

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Expansion of e = 271828182845905 (Euler)

e = 2 +1

1 +1

2 +1

1 +1

1 +1

4 +1

1 +1

1 +1

6 +1

1 +1

1 +1

8 + middot middot middot

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Expansion of π = 314159265358979 (Lange)

π = 3 +12

6 +32

6 +52

6 +72

6 +92

6 + middot middot middot

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

A problematic example

1

2 +1

3 +1

1 +1

minus1 + middot middot middot

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius transformations

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

tn(infin) = 0

Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius transformations

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

tn(infin) = 0

Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius transformations

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

tn(infin) = 0

Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius transformations

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

tn(infin) = 0

Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence again

Convergence of K(an| bn) equivalent to convergence ofT1(0) T2(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Dealing with infin

Previously

1infin

= 0

Now

h(z) =1z

h(infin) = 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Dealing with infin

Previously

1infin

= 0

Now

h(z) =1z

h(infin) = 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps Allows us to bring in geometry Simpler notation composition of maps rather than

algebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps

Allows us to bring in geometry Simpler notation composition of maps rather than

algebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps Allows us to bring in geometry

Simpler notation composition of maps rather thanalgebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps Allows us to bring in geometry Simpler notation composition of maps rather than

algebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Reminder

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Reminder

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions1swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions2swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries

Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group (and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Isometry group

Isom(H3) =M

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Back to continued fractions

tn(infin) = 0 Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Project

The geometry of continued fractions

httpmathsorg ims25mathspresentationsphp

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Continued fractions

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Continued fractions

K(an| bn) =a1

b1 +a2

b2 +a3

b3 +a4

b4 + middot middot middot

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Continued fraction convergence

a1

b1

a1

b1 +a2

b2

a1

b1 +a2

b2 +a3

b3

a1

b1 +a2

b2 +a3

b3 +a4

b4

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Continued fraction convergence

a1

b1

a1

b1 +a2

b2

a1

b1 +a2

b2 +a3

b3

a1

b1 +a2

b2 +a3

b3 +a4

b4

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Continued fraction convergence

a1

b1

a1

b1 +a2

b2

a1

b1 +a2

b2 +a3

b3

a1

b1 +a2

b2 +a3

b3 +a4

b4

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Continued fraction convergence

a1

b1

a1

b1 +a2

b2

a1

b1 +a2

b2 +a3

b3

a1

b1 +a2

b2 +a3

b3 +a4

b4

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Expansion of e = 271828182845905 (Euler)

e = 2 +1

1 +1

2 +1

1 +1

1 +1

4 +1

1 +1

1 +1

6 +1

1 +1

1 +1

8 + middot middot middot

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Expansion of π = 314159265358979 (Lange)

π = 3 +12

6 +32

6 +52

6 +72

6 +92

6 + middot middot middot

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

A problematic example

1

2 +1

3 +1

1 +1

minus1 + middot middot middot

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius transformations

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

tn(infin) = 0

Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius transformations

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

tn(infin) = 0

Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius transformations

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

tn(infin) = 0

Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius transformations

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

tn(infin) = 0

Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence again

Convergence of K(an| bn) equivalent to convergence ofT1(0) T2(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Dealing with infin

Previously

1infin

= 0

Now

h(z) =1z

h(infin) = 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Dealing with infin

Previously

1infin

= 0

Now

h(z) =1z

h(infin) = 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps Allows us to bring in geometry Simpler notation composition of maps rather than

algebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps

Allows us to bring in geometry Simpler notation composition of maps rather than

algebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps Allows us to bring in geometry

Simpler notation composition of maps rather thanalgebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps Allows us to bring in geometry Simpler notation composition of maps rather than

algebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Reminder

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Reminder

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions1swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions2swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries

Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group (and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Isometry group

Isom(H3) =M

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Back to continued fractions

tn(infin) = 0 Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Continued fractions

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Continued fractions

K(an| bn) =a1

b1 +a2

b2 +a3

b3 +a4

b4 + middot middot middot

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Continued fraction convergence

a1

b1

a1

b1 +a2

b2

a1

b1 +a2

b2 +a3

b3

a1

b1 +a2

b2 +a3

b3 +a4

b4

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Continued fraction convergence

a1

b1

a1

b1 +a2

b2

a1

b1 +a2

b2 +a3

b3

a1

b1 +a2

b2 +a3

b3 +a4

b4

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Continued fraction convergence

a1

b1

a1

b1 +a2

b2

a1

b1 +a2

b2 +a3

b3

a1

b1 +a2

b2 +a3

b3 +a4

b4

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Continued fraction convergence

a1

b1

a1

b1 +a2

b2

a1

b1 +a2

b2 +a3

b3

a1

b1 +a2

b2 +a3

b3 +a4

b4

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Expansion of e = 271828182845905 (Euler)

e = 2 +1

1 +1

2 +1

1 +1

1 +1

4 +1

1 +1

1 +1

6 +1

1 +1

1 +1

8 + middot middot middot

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Expansion of π = 314159265358979 (Lange)

π = 3 +12

6 +32

6 +52

6 +72

6 +92

6 + middot middot middot

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

A problematic example

1

2 +1

3 +1

1 +1

minus1 + middot middot middot

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius transformations

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

tn(infin) = 0

Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius transformations

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

tn(infin) = 0

Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius transformations

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

tn(infin) = 0

Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius transformations

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

tn(infin) = 0

Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence again

Convergence of K(an| bn) equivalent to convergence ofT1(0) T2(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Dealing with infin

Previously

1infin

= 0

Now

h(z) =1z

h(infin) = 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Dealing with infin

Previously

1infin

= 0

Now

h(z) =1z

h(infin) = 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps Allows us to bring in geometry Simpler notation composition of maps rather than

algebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps

Allows us to bring in geometry Simpler notation composition of maps rather than

algebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps Allows us to bring in geometry

Simpler notation composition of maps rather thanalgebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps Allows us to bring in geometry Simpler notation composition of maps rather than

algebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Reminder

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Reminder

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions1swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions2swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries

Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group (and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Isometry group

Isom(H3) =M

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Back to continued fractions

tn(infin) = 0 Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Continued fractions

K(an| bn) =a1

b1 +a2

b2 +a3

b3 +a4

b4 + middot middot middot

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Continued fraction convergence

a1

b1

a1

b1 +a2

b2

a1

b1 +a2

b2 +a3

b3

a1

b1 +a2

b2 +a3

b3 +a4

b4

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Continued fraction convergence

a1

b1

a1

b1 +a2

b2

a1

b1 +a2

b2 +a3

b3

a1

b1 +a2

b2 +a3

b3 +a4

b4

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Continued fraction convergence

a1

b1

a1

b1 +a2

b2

a1

b1 +a2

b2 +a3

b3

a1

b1 +a2

b2 +a3

b3 +a4

b4

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Continued fraction convergence

a1

b1

a1

b1 +a2

b2

a1

b1 +a2

b2 +a3

b3

a1

b1 +a2

b2 +a3

b3 +a4

b4

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Expansion of e = 271828182845905 (Euler)

e = 2 +1

1 +1

2 +1

1 +1

1 +1

4 +1

1 +1

1 +1

6 +1

1 +1

1 +1

8 + middot middot middot

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Expansion of π = 314159265358979 (Lange)

π = 3 +12

6 +32

6 +52

6 +72

6 +92

6 + middot middot middot

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

A problematic example

1

2 +1

3 +1

1 +1

minus1 + middot middot middot

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius transformations

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

tn(infin) = 0

Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius transformations

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

tn(infin) = 0

Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius transformations

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

tn(infin) = 0

Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius transformations

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

tn(infin) = 0

Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence again

Convergence of K(an| bn) equivalent to convergence ofT1(0) T2(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Dealing with infin

Previously

1infin

= 0

Now

h(z) =1z

h(infin) = 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Dealing with infin

Previously

1infin

= 0

Now

h(z) =1z

h(infin) = 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps Allows us to bring in geometry Simpler notation composition of maps rather than

algebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps

Allows us to bring in geometry Simpler notation composition of maps rather than

algebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps Allows us to bring in geometry

Simpler notation composition of maps rather thanalgebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps Allows us to bring in geometry Simpler notation composition of maps rather than

algebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Reminder

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Reminder

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions1swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions2swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries

Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group (and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Isometry group

Isom(H3) =M

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Back to continued fractions

tn(infin) = 0 Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Continued fraction convergence

a1

b1

a1

b1 +a2

b2

a1

b1 +a2

b2 +a3

b3

a1

b1 +a2

b2 +a3

b3 +a4

b4

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Continued fraction convergence

a1

b1

a1

b1 +a2

b2

a1

b1 +a2

b2 +a3

b3

a1

b1 +a2

b2 +a3

b3 +a4

b4

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Continued fraction convergence

a1

b1

a1

b1 +a2

b2

a1

b1 +a2

b2 +a3

b3

a1

b1 +a2

b2 +a3

b3 +a4

b4

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Continued fraction convergence

a1

b1

a1

b1 +a2

b2

a1

b1 +a2

b2 +a3

b3

a1

b1 +a2

b2 +a3

b3 +a4

b4

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Expansion of e = 271828182845905 (Euler)

e = 2 +1

1 +1

2 +1

1 +1

1 +1

4 +1

1 +1

1 +1

6 +1

1 +1

1 +1

8 + middot middot middot

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Expansion of π = 314159265358979 (Lange)

π = 3 +12

6 +32

6 +52

6 +72

6 +92

6 + middot middot middot

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

A problematic example

1

2 +1

3 +1

1 +1

minus1 + middot middot middot

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius transformations

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

tn(infin) = 0

Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius transformations

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

tn(infin) = 0

Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius transformations

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

tn(infin) = 0

Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius transformations

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

tn(infin) = 0

Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence again

Convergence of K(an| bn) equivalent to convergence ofT1(0) T2(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Dealing with infin

Previously

1infin

= 0

Now

h(z) =1z

h(infin) = 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Dealing with infin

Previously

1infin

= 0

Now

h(z) =1z

h(infin) = 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps Allows us to bring in geometry Simpler notation composition of maps rather than

algebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps

Allows us to bring in geometry Simpler notation composition of maps rather than

algebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps Allows us to bring in geometry

Simpler notation composition of maps rather thanalgebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps Allows us to bring in geometry Simpler notation composition of maps rather than

algebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Reminder

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Reminder

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions1swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions2swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries

Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group (and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Isometry group

Isom(H3) =M

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Back to continued fractions

tn(infin) = 0 Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Continued fraction convergence

a1

b1

a1

b1 +a2

b2

a1

b1 +a2

b2 +a3

b3

a1

b1 +a2

b2 +a3

b3 +a4

b4

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Continued fraction convergence

a1

b1

a1

b1 +a2

b2

a1

b1 +a2

b2 +a3

b3

a1

b1 +a2

b2 +a3

b3 +a4

b4

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Continued fraction convergence

a1

b1

a1

b1 +a2

b2

a1

b1 +a2

b2 +a3

b3

a1

b1 +a2

b2 +a3

b3 +a4

b4

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Expansion of e = 271828182845905 (Euler)

e = 2 +1

1 +1

2 +1

1 +1

1 +1

4 +1

1 +1

1 +1

6 +1

1 +1

1 +1

8 + middot middot middot

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Expansion of π = 314159265358979 (Lange)

π = 3 +12

6 +32

6 +52

6 +72

6 +92

6 + middot middot middot

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

A problematic example

1

2 +1

3 +1

1 +1

minus1 + middot middot middot

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius transformations

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

tn(infin) = 0

Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius transformations

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

tn(infin) = 0

Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius transformations

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

tn(infin) = 0

Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius transformations

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

tn(infin) = 0

Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence again

Convergence of K(an| bn) equivalent to convergence ofT1(0) T2(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Dealing with infin

Previously

1infin

= 0

Now

h(z) =1z

h(infin) = 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Dealing with infin

Previously

1infin

= 0

Now

h(z) =1z

h(infin) = 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps Allows us to bring in geometry Simpler notation composition of maps rather than

algebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps

Allows us to bring in geometry Simpler notation composition of maps rather than

algebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps Allows us to bring in geometry

Simpler notation composition of maps rather thanalgebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps Allows us to bring in geometry Simpler notation composition of maps rather than

algebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Reminder

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Reminder

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions1swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions2swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries

Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group (and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Isometry group

Isom(H3) =M

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Back to continued fractions

tn(infin) = 0 Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Continued fraction convergence

a1

b1

a1

b1 +a2

b2

a1

b1 +a2

b2 +a3

b3

a1

b1 +a2

b2 +a3

b3 +a4

b4

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Continued fraction convergence

a1

b1

a1

b1 +a2

b2

a1

b1 +a2

b2 +a3

b3

a1

b1 +a2

b2 +a3

b3 +a4

b4

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Expansion of e = 271828182845905 (Euler)

e = 2 +1

1 +1

2 +1

1 +1

1 +1

4 +1

1 +1

1 +1

6 +1

1 +1

1 +1

8 + middot middot middot

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Expansion of π = 314159265358979 (Lange)

π = 3 +12

6 +32

6 +52

6 +72

6 +92

6 + middot middot middot

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

A problematic example

1

2 +1

3 +1

1 +1

minus1 + middot middot middot

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius transformations

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

tn(infin) = 0

Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius transformations

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

tn(infin) = 0

Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius transformations

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

tn(infin) = 0

Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius transformations

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

tn(infin) = 0

Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence again

Convergence of K(an| bn) equivalent to convergence ofT1(0) T2(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Dealing with infin

Previously

1infin

= 0

Now

h(z) =1z

h(infin) = 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Dealing with infin

Previously

1infin

= 0

Now

h(z) =1z

h(infin) = 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps Allows us to bring in geometry Simpler notation composition of maps rather than

algebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps

Allows us to bring in geometry Simpler notation composition of maps rather than

algebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps Allows us to bring in geometry

Simpler notation composition of maps rather thanalgebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps Allows us to bring in geometry Simpler notation composition of maps rather than

algebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Reminder

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Reminder

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions1swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions2swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries

Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group (and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Isometry group

Isom(H3) =M

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Back to continued fractions

tn(infin) = 0 Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Continued fraction convergence

a1

b1

a1

b1 +a2

b2

a1

b1 +a2

b2 +a3

b3

a1

b1 +a2

b2 +a3

b3 +a4

b4

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Expansion of e = 271828182845905 (Euler)

e = 2 +1

1 +1

2 +1

1 +1

1 +1

4 +1

1 +1

1 +1

6 +1

1 +1

1 +1

8 + middot middot middot

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Expansion of π = 314159265358979 (Lange)

π = 3 +12

6 +32

6 +52

6 +72

6 +92

6 + middot middot middot

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

A problematic example

1

2 +1

3 +1

1 +1

minus1 + middot middot middot

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius transformations

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

tn(infin) = 0

Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius transformations

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

tn(infin) = 0

Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius transformations

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

tn(infin) = 0

Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius transformations

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

tn(infin) = 0

Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence again

Convergence of K(an| bn) equivalent to convergence ofT1(0) T2(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Dealing with infin

Previously

1infin

= 0

Now

h(z) =1z

h(infin) = 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Dealing with infin

Previously

1infin

= 0

Now

h(z) =1z

h(infin) = 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps Allows us to bring in geometry Simpler notation composition of maps rather than

algebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps

Allows us to bring in geometry Simpler notation composition of maps rather than

algebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps Allows us to bring in geometry

Simpler notation composition of maps rather thanalgebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps Allows us to bring in geometry Simpler notation composition of maps rather than

algebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Reminder

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Reminder

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions1swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions2swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries

Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group (and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Isometry group

Isom(H3) =M

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Back to continued fractions

tn(infin) = 0 Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Expansion of e = 271828182845905 (Euler)

e = 2 +1

1 +1

2 +1

1 +1

1 +1

4 +1

1 +1

1 +1

6 +1

1 +1

1 +1

8 + middot middot middot

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Expansion of π = 314159265358979 (Lange)

π = 3 +12

6 +32

6 +52

6 +72

6 +92

6 + middot middot middot

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

A problematic example

1

2 +1

3 +1

1 +1

minus1 + middot middot middot

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius transformations

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

tn(infin) = 0

Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius transformations

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

tn(infin) = 0

Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius transformations

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

tn(infin) = 0

Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius transformations

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

tn(infin) = 0

Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence again

Convergence of K(an| bn) equivalent to convergence ofT1(0) T2(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Dealing with infin

Previously

1infin

= 0

Now

h(z) =1z

h(infin) = 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Dealing with infin

Previously

1infin

= 0

Now

h(z) =1z

h(infin) = 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps Allows us to bring in geometry Simpler notation composition of maps rather than

algebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps

Allows us to bring in geometry Simpler notation composition of maps rather than

algebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps Allows us to bring in geometry

Simpler notation composition of maps rather thanalgebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps Allows us to bring in geometry Simpler notation composition of maps rather than

algebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Reminder

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Reminder

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions1swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions2swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries

Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group (and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Isometry group

Isom(H3) =M

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Back to continued fractions

tn(infin) = 0 Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Expansion of π = 314159265358979 (Lange)

π = 3 +12

6 +32

6 +52

6 +72

6 +92

6 + middot middot middot

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

A problematic example

1

2 +1

3 +1

1 +1

minus1 + middot middot middot

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius transformations

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

tn(infin) = 0

Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius transformations

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

tn(infin) = 0

Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius transformations

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

tn(infin) = 0

Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius transformations

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

tn(infin) = 0

Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence again

Convergence of K(an| bn) equivalent to convergence ofT1(0) T2(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Dealing with infin

Previously

1infin

= 0

Now

h(z) =1z

h(infin) = 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Dealing with infin

Previously

1infin

= 0

Now

h(z) =1z

h(infin) = 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps Allows us to bring in geometry Simpler notation composition of maps rather than

algebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps

Allows us to bring in geometry Simpler notation composition of maps rather than

algebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps Allows us to bring in geometry

Simpler notation composition of maps rather thanalgebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps Allows us to bring in geometry Simpler notation composition of maps rather than

algebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Reminder

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Reminder

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions1swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions2swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries

Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group (and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Isometry group

Isom(H3) =M

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Back to continued fractions

tn(infin) = 0 Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

A problematic example

1

2 +1

3 +1

1 +1

minus1 + middot middot middot

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius transformations

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

tn(infin) = 0

Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius transformations

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

tn(infin) = 0

Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius transformations

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

tn(infin) = 0

Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius transformations

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

tn(infin) = 0

Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence again

Convergence of K(an| bn) equivalent to convergence ofT1(0) T2(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Dealing with infin

Previously

1infin

= 0

Now

h(z) =1z

h(infin) = 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Dealing with infin

Previously

1infin

= 0

Now

h(z) =1z

h(infin) = 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps Allows us to bring in geometry Simpler notation composition of maps rather than

algebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps

Allows us to bring in geometry Simpler notation composition of maps rather than

algebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps Allows us to bring in geometry

Simpler notation composition of maps rather thanalgebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps Allows us to bring in geometry Simpler notation composition of maps rather than

algebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Reminder

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Reminder

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions1swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions2swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries

Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group (and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Isometry group

Isom(H3) =M

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Back to continued fractions

tn(infin) = 0 Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius transformations

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

tn(infin) = 0

Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius transformations

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

tn(infin) = 0

Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius transformations

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

tn(infin) = 0

Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius transformations

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

tn(infin) = 0

Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence again

Convergence of K(an| bn) equivalent to convergence ofT1(0) T2(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Dealing with infin

Previously

1infin

= 0

Now

h(z) =1z

h(infin) = 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Dealing with infin

Previously

1infin

= 0

Now

h(z) =1z

h(infin) = 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps Allows us to bring in geometry Simpler notation composition of maps rather than

algebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps

Allows us to bring in geometry Simpler notation composition of maps rather than

algebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps Allows us to bring in geometry

Simpler notation composition of maps rather thanalgebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps Allows us to bring in geometry Simpler notation composition of maps rather than

algebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Reminder

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Reminder

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions1swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions2swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries

Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group (and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Isometry group

Isom(H3) =M

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Back to continued fractions

tn(infin) = 0 Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius transformations

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

tn(infin) = 0

Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius transformations

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

tn(infin) = 0

Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius transformations

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

tn(infin) = 0

Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence again

Convergence of K(an| bn) equivalent to convergence ofT1(0) T2(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Dealing with infin

Previously

1infin

= 0

Now

h(z) =1z

h(infin) = 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Dealing with infin

Previously

1infin

= 0

Now

h(z) =1z

h(infin) = 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps Allows us to bring in geometry Simpler notation composition of maps rather than

algebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps

Allows us to bring in geometry Simpler notation composition of maps rather than

algebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps Allows us to bring in geometry

Simpler notation composition of maps rather thanalgebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps Allows us to bring in geometry Simpler notation composition of maps rather than

algebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Reminder

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Reminder

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions1swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions2swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries

Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group (and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Isometry group

Isom(H3) =M

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Back to continued fractions

tn(infin) = 0 Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius transformations

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

tn(infin) = 0

Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius transformations

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

tn(infin) = 0

Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence again

Convergence of K(an| bn) equivalent to convergence ofT1(0) T2(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Dealing with infin

Previously

1infin

= 0

Now

h(z) =1z

h(infin) = 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Dealing with infin

Previously

1infin

= 0

Now

h(z) =1z

h(infin) = 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps Allows us to bring in geometry Simpler notation composition of maps rather than

algebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps

Allows us to bring in geometry Simpler notation composition of maps rather than

algebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps Allows us to bring in geometry

Simpler notation composition of maps rather thanalgebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps Allows us to bring in geometry Simpler notation composition of maps rather than

algebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Reminder

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Reminder

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions1swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions2swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries

Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group (and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Isometry group

Isom(H3) =M

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Back to continued fractions

tn(infin) = 0 Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius transformations

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

tn(infin) = 0

Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence again

Convergence of K(an| bn) equivalent to convergence ofT1(0) T2(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Dealing with infin

Previously

1infin

= 0

Now

h(z) =1z

h(infin) = 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Dealing with infin

Previously

1infin

= 0

Now

h(z) =1z

h(infin) = 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps Allows us to bring in geometry Simpler notation composition of maps rather than

algebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps

Allows us to bring in geometry Simpler notation composition of maps rather than

algebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps Allows us to bring in geometry

Simpler notation composition of maps rather thanalgebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps Allows us to bring in geometry Simpler notation composition of maps rather than

algebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Reminder

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Reminder

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions1swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions2swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries

Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group (and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Isometry group

Isom(H3) =M

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Back to continued fractions

tn(infin) = 0 Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence again

Convergence of K(an| bn) equivalent to convergence ofT1(0) T2(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Dealing with infin

Previously

1infin

= 0

Now

h(z) =1z

h(infin) = 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Dealing with infin

Previously

1infin

= 0

Now

h(z) =1z

h(infin) = 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps Allows us to bring in geometry Simpler notation composition of maps rather than

algebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps

Allows us to bring in geometry Simpler notation composition of maps rather than

algebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps Allows us to bring in geometry

Simpler notation composition of maps rather thanalgebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps Allows us to bring in geometry Simpler notation composition of maps rather than

algebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Reminder

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Reminder

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions1swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions2swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries

Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group (and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Isometry group

Isom(H3) =M

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Back to continued fractions

tn(infin) = 0 Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Dealing with infin

Previously

1infin

= 0

Now

h(z) =1z

h(infin) = 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Dealing with infin

Previously

1infin

= 0

Now

h(z) =1z

h(infin) = 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps Allows us to bring in geometry Simpler notation composition of maps rather than

algebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps

Allows us to bring in geometry Simpler notation composition of maps rather than

algebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps Allows us to bring in geometry

Simpler notation composition of maps rather thanalgebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps Allows us to bring in geometry Simpler notation composition of maps rather than

algebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Reminder

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Reminder

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions1swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions2swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries

Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group (and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Isometry group

Isom(H3) =M

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Back to continued fractions

tn(infin) = 0 Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Dealing with infin

Previously

1infin

= 0

Now

h(z) =1z

h(infin) = 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps Allows us to bring in geometry Simpler notation composition of maps rather than

algebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps

Allows us to bring in geometry Simpler notation composition of maps rather than

algebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps Allows us to bring in geometry

Simpler notation composition of maps rather thanalgebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps Allows us to bring in geometry Simpler notation composition of maps rather than

algebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Reminder

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Reminder

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions1swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions2swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries

Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group (and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Isometry group

Isom(H3) =M

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Back to continued fractions

tn(infin) = 0 Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps Allows us to bring in geometry Simpler notation composition of maps rather than

algebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps

Allows us to bring in geometry Simpler notation composition of maps rather than

algebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps Allows us to bring in geometry

Simpler notation composition of maps rather thanalgebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps Allows us to bring in geometry Simpler notation composition of maps rather than

algebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Reminder

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Reminder

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions1swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions2swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries

Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group (and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Isometry group

Isom(H3) =M

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Back to continued fractions

tn(infin) = 0 Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps

Allows us to bring in geometry Simpler notation composition of maps rather than

algebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps Allows us to bring in geometry

Simpler notation composition of maps rather thanalgebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps Allows us to bring in geometry Simpler notation composition of maps rather than

algebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Reminder

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Reminder

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions1swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions2swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries

Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group (and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Isometry group

Isom(H3) =M

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Back to continued fractions

tn(infin) = 0 Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps Allows us to bring in geometry

Simpler notation composition of maps rather thanalgebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps Allows us to bring in geometry Simpler notation composition of maps rather than

algebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Reminder

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Reminder

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions1swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions2swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries

Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group (and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Isometry group

Isom(H3) =M

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Back to continued fractions

tn(infin) = 0 Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three more reasons for using Mobius maps

There is already a well developed theory of Mobius maps Allows us to bring in geometry Simpler notation composition of maps rather than

algebraic manipulation

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Reminder

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Reminder

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions1swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions2swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries

Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group (and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Isometry group

Isom(H3) =M

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Back to continued fractions

tn(infin) = 0 Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Reminder

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Reminder

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions1swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions2swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries

Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group (and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Isometry group

Isom(H3) =M

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Back to continued fractions

tn(infin) = 0 Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Reminder

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Reminder

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions1swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions2swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries

Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group (and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Isometry group

Isom(H3) =M

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Back to continued fractions

tn(infin) = 0 Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Reminder

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Reminder

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions1swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions2swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries

Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group (and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Isometry group

Isom(H3) =M

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Back to continued fractions

tn(infin) = 0 Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Reminder

tn(z) =an

bn + z

Tn = t1 t2 middot middot middot tn

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions1swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions2swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries

Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group (and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Isometry group

Isom(H3) =M

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Back to continued fractions

tn(infin) = 0 Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions1swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions2swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries

Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group (and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Isometry group

Isom(H3) =M

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Back to continued fractions

tn(infin) = 0 Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions1swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions2swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries

Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group (and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Isometry group

Isom(H3) =M

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Back to continued fractions

tn(infin) = 0 Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions1swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions2swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries

Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group (and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Isometry group

Isom(H3) =M

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Back to continued fractions

tn(infin) = 0 Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Usual integer continued fractions

K(1| bn) bn isin N

tn(z) =1

bn + z

The modular group

Γ =z 7rarr az + b

cz + d a b c d isin Z adminus bc = 1

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions1swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions2swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries

Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group (and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Isometry group

Isom(H3) =M

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Back to continued fractions

tn(infin) = 0 Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions1swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions2swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries

Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group (and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Isometry group

Isom(H3) =M

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Back to continued fractions

tn(infin) = 0 Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions1swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions2swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries

Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group (and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Isometry group

Isom(H3) =M

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Back to continued fractions

tn(infin) = 0 Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions1swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions2swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries

Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group (and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Isometry group

Isom(H3) =M

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Back to continued fractions

tn(infin) = 0 Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Complex continued fractions

K(an| bn) an bn isin C

tn(z) =an

bn + z

The Mobius group

M =z 7rarr az + b

cz + d a b c d isin C adminus bc 6= 0

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions1swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions2swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries

Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group (and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Isometry group

Isom(H3) =M

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Back to continued fractions

tn(infin) = 0 Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions1swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions2swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries

Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group (and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Isometry group

Isom(H3) =M

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Back to continued fractions

tn(infin) = 0 Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

M generated by inversions

Inversions2swf
Media File (applicationx-shockwave-flash)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries

Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group (and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Isometry group

Isom(H3) =M

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Back to continued fractions

tn(infin) = 0 Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries

Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group (and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Isometry group

Isom(H3) =M

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Back to continued fractions

tn(infin) = 0 Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries

Topological group

(and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group (and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Isometry group

Isom(H3) =M

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Back to continued fractions

tn(infin) = 0 Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Two aspects of M

Three-dimensional hyperbolic isometries Topological group (and complete metric space)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Isometry group

Isom(H3) =M

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Back to continued fractions

tn(infin) = 0 Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Isometry group

Isom(H3) =M

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Back to continued fractions

tn(infin) = 0 Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Isometry group

Isom(H3) =M

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Back to continued fractions

tn(infin) = 0 Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Isometry group

Isom(H3) =M

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Back to continued fractions

tn(infin) = 0 Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Isometry group

Isom(H3) =M

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Back to continued fractions

tn(infin) = 0 Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem I

Interpret each result on complex continued fractions in terms of(a) hyperbolic geometry and (b) topological group theory

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Isometry group

Isom(H3) =M

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Back to continued fractions

tn(infin) = 0 Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Isometry group

Isom(H3) =M

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Back to continued fractions

tn(infin) = 0 Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

hyperbolic geometry

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Isometry group

Isom(H3) =M

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Back to continued fractions

tn(infin) = 0 Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Isometry group

Isom(H3) =M

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Back to continued fractions

tn(infin) = 0 Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

H3 = (x1 x2 x3) isin R3 x3 gt 0

Clarrrarr (x1 x2 0) isin R3 x1 x2 isin R

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Isometry group

Isom(H3) =M

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Back to continued fractions

tn(infin) = 0 Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Upper half-space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Isometry group

Isom(H3) =M

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Back to continued fractions

tn(infin) = 0 Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Isometry group

Isom(H3) =M

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Back to continued fractions

tn(infin) = 0 Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

(H3 ρ)

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Isometry group

Isom(H3) =M

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Back to continued fractions

tn(infin) = 0 Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Isometry group

Isom(H3) =M

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Back to continued fractions

tn(infin) = 0 Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Three-dimensional hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Isometry group

Isom(H3) =M

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Back to continued fractions

tn(infin) = 0 Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Isometry group

Isom(H3) =M

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Back to continued fractions

tn(infin) = 0 Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Isometry group

Isom(H3) =M

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Back to continued fractions

tn(infin) = 0 Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Graph representation of hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Isometry group

Isom(H3) =M

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Back to continued fractions

tn(infin) = 0 Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Isometry group

Isom(H3) =M

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Back to continued fractions

tn(infin) = 0 Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius action on hyperbolic space

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Isometry group

Isom(H3) =M

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Back to continued fractions

tn(infin) = 0 Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Isometry group

Isom(H3) =M

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Back to continued fractions

tn(infin) = 0 Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Back to continued fractions

tn(infin) = 0 Tn(infin) = Tnminus1(0)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Suppose K(an| bn) converges

In other words suppose T1(0) T2(0) converges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Convergence

IfTn(0)rarr p

thenTn(j)rarr p

(Lorentzen Aebischer Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the hyperbolic metric

sinh 12ρ(x y) =

|xminus y|2radicx3y3

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j))

= ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))

= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

ρ(j tn(j)) = ρ(h(j) htn(j))= ρ(j bn + j)

sinh 12ρ(j tn(j)) =

|bn|2

ρ(j tn(j)) = ρ(Tnminus1(j) Tn(j))

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin then

sumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Hyperbolic distance calculations

sinh 12ρ(Tnminus1(j) Tn(j)) =

|bn|2

Hence ifsum

n |bn| lt +infin thensumn

ρ(Tnminus1(j) Tn(j)) lt +infin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Cannot reach the boundary (Beardon)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Open Problem II

What is the geometric significance of the argument of bn to theorbit T1(j) T2(j)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Schematic diagram

continued fractions

Mobius maps

topological groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3

Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

Group of hyperbolic isometries of H3 Group of conformal automorphisms of Cinfin

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Stereographic projection

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

χ(w z) =2|w minus z|radic

1 + |w|2radic

1 + |z|2χ(winfin) =

2radic1 + |w|2

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The chordal metric

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

The supremum metric

χ0(f g) = supzisinCinfin

χ(f(z) g(z))

f g isinM

The metric of uniform convergence

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space

(M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group

right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g)

h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Mobius group

(M χ0) is a complete metric space (M χ0) is a topological group right-invariant χ0(fk gk) = χ0(f g) h(z) = 1z is a chordal isometry χ0(hf hg) = χ0(f g)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Recall the SternndashStolz Theorem

Theorem Ifsum

n |bn| converges then K(1| bn) diverges

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Key observation

If bn small then

tn(z) =1

bn + zsim h(z) =

1z

We must calculate χ0(tn h)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h)

= χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)

= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(tn h)

χ0(tn h) = χ0(htn I)= χ0(z + bn z)

= supzisinCinfin

2|bn|radic1 + |z|2

radic1 + |z + bn|2

le 2|bn|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2)

= χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)

= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)

le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)

= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)

le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Calculate χ0(Tminus1n Tminus1

n+2)

χ0(Tminus1n Tminus1

n+2) = χ0(tn+1tn+2 I)= χ0(htn+1tn+2 h)le χ0(htn+1tn+2 tn+2) + χ0(tn+2 h)= χ0(tn+1 h) + χ0(tn+2 h)le 2|bn+1|+ 2|bn+2|

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin then

sumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

Ifsum

n |bn| lt +infin thensumn

χ0(Tminus1n Tminus1

n+2) lt +infin

Hence Tminus12nminus1 converges uniformly to a Mobius map f

Hence T2nminus1 converges uniformly to a Mobius map g

Hence T2n = T2nminus1t2n converges uniformly to gh whereh(z) = 1z

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Conclusion

In summary ifsum

n |bn| lt +infin then there is a Mobius map gsuch that

T2nminus1 rarr g T2n rarr gh

Hence

T2nminus1(0)rarr g(0) T2n(0)rarr gh(0) = g(infin)

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups

Continued fractions Mobius maps Hyperbolic geometry Toplogical groups

Thank you

  • Continued fractions
    • Moumlbius maps
      • Hyperbolic geometry
        • Toplogical groups

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