general theory of unitary pisot substitution · applications low complexity and sturmian sequences...

Background: Low complexityFree groups automorphisms and tilings: An example

Applications

Low complexity and sturmian sequencesDiscrete lines in the planeGeneralizations: Hyperplanes, lines and Pisot substitutionsGeneralizations

General theory of unitary Pisot substitution

I The case of Pisot substitution in now well understood.

I Crowning the work of several people, a recent paper (Bargeand Kwapisz) shows that, if the substitution satisfies theso-called ”strong coincidence condition”, the structure of theassociated system is completely understood:

I discrete lines and planes

I Rauzy fractal

I symbolic dynamics

I All known Pisot substitutions satisfy the strong coincidencecondition

Pierre Arnoux Pavages, substitutions et automorphismes de groupe

Applications

I Rauzy fractal

I symbolic dynamics

Applications

I Rauzy fractal

I symbolic dynamics

Applications

I Rauzy fractal

I symbolic dynamics

Applications

I Rauzy fractal

I symbolic dynamics

Applications

I Rauzy fractal

I symbolic dynamics

Applications

Discrete 2-planes in R4

Applications

Discrete 2-planes in R4

Applications

Possible generalizations

Unitary Pisot substitutions

I Non-unitary substitutions:p-adic component (Siegel)

I From substitutions to Pisotautomorphisms of free groups:recent results(A-Berthe-Hilion-Siegel)

I Non Pisot case: the firstexample

Applications

A remark on entropy

I The entropy of a toral automorphism is the sum of thelogarithms of the eigenvalues larger than 1.

I The entropy of the subshift of finite type associated to asubstitution is the logarithm of the largest eigenvalue.

I These coincide only in the Pisot case

I This hints to the necessity of considering exterior powers ofthe matrix A

Applications

A remark on entropy

Applications

A remark on entropy

Applications

A remark on entropy

Applications

NotationsExtensions of free group automorphismsPlane tiling

A free group automorphism

σ automorphism of the free group F4:

1 7→ 2

2 7→ 3

3 7→ 4

4 7→ 41−1

Matrix M =

0 0 0 −11 0 0 00 1 0 00 0 1 1

Applications

Characteristic polynomial X 4 − X 3 + 1Eigenvalues 1.01891± 0.602565i , −0.518913± 0.66661iNon Pisot!Expanding plane Pe ≡ CContracting plane Pc ≡ CAssociated projections πe , πc

Applications

Projections of the canonical basis

-0.2 0.2 0.4 0.6 0.8 1 1.2

!e!e1"

!e!e2"

!e!e3"!e!e4"

-0.5-0.25 0.25 0.5 0.75 1

!c!e1"

!c!e2"

!c!e3"

!c!e4"

Applications

Geometric extensions of free group automorphisms I

I Words as discrete lines in R4

I σ acts naturally on discrete lines in R4

I Map E1(σ) defined on space G1 of weighted sum of discretelines

I This is still 1-dimensional!

Applications

Geometric extensions of free group automorphisms II

I i : s 7→ i(s) and j : t 7→ j(t)segments; define the orientedface i ∧ j as the oriented surface(s, t) 7→ i(s) + j(t).

I σ acts in a natural way on facesby taking i ∧ j toE1(σ)(i) ∧ E1(σ)(j)

I Map E2(σ) defined on space G2

of weighted sum of discretefaces

I The matrix of E2(σ) is positive!

πe (0, 1 ∧ 2)

πe (0, 1 ∧ 3)

πe (0, 1 ∧ 4)

πe (0, 2 ∧ 3)

πe (0, 2 ∧ 4)

πe (0, 3 ∧ 4)

Applications

πe (0, 2 ∧ 4)

E2(σ)→

πe (E2(σ)(0, 2 ∧ 4))

Applications

πe (0, 1 ∧ 2)

E2(σ)→

πe (0, 2 ∧ 3)

πe (0, 1 ∧ 3)

E2(σ)→

πe (0, 2 ∧ 4)

πe (0, 1 ∧ 4)

E2(σ)→

πe (0, 2 ∧ 4) +πe (−e1 − e4, 1 ∧ 2)

πe (0, 2 ∧ 3)

E2(σ)→

πe (0, 3 ∧ 4)

πe (0, 2 ∧ 4)

E2(σ)→

πe (0, 3 ∧ 4) +πe (−e1 + e4, 1 ∧ 3)

πe (0, 3 ∧ 4)

E2(σ)→

πe (−e1 + e4, 1 ∧ 4)

Applications

πe (0, 1 ∧ 2)

E2(σ)→

πe (0, 2 ∧ 3)

πe (0, 1 ∧ 3)

E2(σ)→

πe (0, 2 ∧ 4)

πe (0, 1 ∧ 4)

E2(σ)→

πe (0, 2 ∧ 4) +πe (−e1 − e4, 1 ∧ 2)

πe (0, 2 ∧ 3)

E2(σ)→

πe (0, 3 ∧ 4)

πe (0, 2 ∧ 4)

E2(σ)→

πe (0, 3 ∧ 4) +πe (−e1 + e4, 1 ∧ 3)

πe (0, 3 ∧ 4)

E2(σ)→

πe (−e1 + e4, 1 ∧ 4)

Applications

Geometric extensions III: formalism

I We define the space of formal finite sums of weighted 2-faces(x, i ∧ j), with x ∈ Z4.

I The 2-dimensional extension E2(σ) is defined on this space by:E2(σ)(x, i ∧ j) :=∑li

∑ljn=1

(A(x) + f (P

(i)m ) + f (P

(j)m ), W

(i)m ∧W

)I The matrix associated to E2(σ) is the exterior square of M:

0 0 0 1 0 00 0 0 0 1 01 0 0 0 0 00 0 0 0 0 10 1 0 1 0 00 0 1 0 1 0

Applications

∑ljn=1

(A(x) + f (P

(i)m ) + f (P

(j)m ), W

(i)m ∧W

0 0 0 1 0 00 0 0 0 1 01 0 0 0 0 00 0 0 0 0 10 1 0 1 0 00 0 1 0 1 0

Applications

∑ljn=1

(A(x) + f (P

(i)m ) + f (P

(j)m ), W

(i)m ∧W

0 0 0 1 0 00 0 0 0 1 01 0 0 0 0 00 0 0 0 0 10 1 0 1 0 00 0 1 0 1 0

Applications

I The matrix associated to E2(σ) is the exterior square of M:

0 0 0 1 0 00 0 0 0 1 01 0 0 0 0 00 0 0 0 0 10 1 0 1 0 00 0 1 0 1 0

I This matrix is positive!

I This is the correct generalization of the notion of substitution(seen as a positive free group endomorphism) in the non-Pisotcase

Applications

0 0 0 1 0 00 0 0 0 1 01 0 0 0 0 00 0 0 0 0 10 1 0 1 0 00 0 1 0 1 0

Applications

0 0 0 1 0 00 0 0 0 1 01 0 0 0 0 00 0 0 0 0 10 1 0 1 0 00 0 1 0 1 0

Applications

A new substitution tiling of the plane

I by projection πe :

I A substitution rule on theexpanding plane

I That generates asubstitution polygonaltiling

Applications

πe (0, 1 ∧ 2)

E2(σ)→

πe (0, 2 ∧ 3)

πe (0, 1 ∧ 3)

E2(σ)→

πe (0, 2 ∧ 4)

πe (0, 1 ∧ 4)

E2(σ)→

πe (0, 2 ∧ 4) +πe (−e1 − e4, 1 ∧ 2)

πe (0, 2 ∧ 3)

E2(σ)→

πe (0, 3 ∧ 4)

πe (0, 2 ∧ 4)

E2(σ)→

πe (0, 3 ∧ 4) +πe (−e1 + e4, 1 ∧ 3)

πe (0, 3 ∧ 4)

E2(σ)→

πe (−e1 + e4, 1 ∧ 4)

Applications

49 step

Πee1Πee2Πee3

412 step

Πee1Πee2Πee3

421 step

Πee1Πee2Πee3

46 step

Πee1Πee2Πee3

47 step

Πee2Πee3

48 step

Πee1Πee2Πee3

43 step

Πee1Πee2Πee3

44 step

Πee2Πee3

45 step

Πee2Πee3

40 step

Πee1Πee2Πee3

41 step

Πee1Πee2Πee3

42 step

Πee1Πee2Πee3

Applications

An exact substitution tiling

I By replacing each face by the limit of its renormalization, oneobtains an exactly self-similar tiling, with fractal tiles.

I The fractal tiles are solutions of a graph-directed IFS given bythe substitution rule.

Applications

An exact substitution tiling

I By replacing each face by the limit of its renormalization, oneobtains an exactly self-similar tiling, with fractal tiles.

I The fractal tiles are solutions of a graph-directed IFS given bythe substitution rule.

Applications

5/09/06 1:40Nautilus_fract_patch_03.gif 932x738 pixels

sur 1file://localhost/Users/pierrearnoux/Desktop/Nautilus_fract_patch_03.gif

Applications

A discrete surface in R4

I The tiling lifts to a unique discrete surface in R4

I Discrete approximation of the expanding plane

Applications

A discrete surface in R4

I The tiling lifts to a unique discrete surface in R4

I Discrete approximation of the expanding plane

Applications

The discrete surface

Applications

Duality

We can do exactly the same for the contracting plane:Define the dual map E 2(σ).It is also positive.Get dual substitution tiling and a dual self-similar tiling.

Applications

5/09/06 1:41Conch_fract_patch_03.gif 926x734 pixels

sur 1file://localhost/Users/pierrearnoux/Desktop/Conch_fract_patch_03.gif

Applications

Quasi-crystals and Rauzy fractalsA complex number systemSymbolic dynamicsOpen problems

Cut-and-project tiling

I The fractal tiles of the expanding tiling are solution of a GIFS.

I The vertices of the contracting tiling are solution of a GIFS.

I After projection on the expanding space, we can observe avery much curious phenomenon:

I The second IFS is the opposite of the first!

I These polygonal tilings are cut-and-project tilings.

Applications

Generalized Rauzy fractals

The window for the tiling of the expanding plane is the contractingRauzy fractal X c = ∪X c

i∧j .It can be obtained by projecting on the contracting plane thevertices of the discrete approximation to the expanding plane.It can also be obtained by renormalization of the projection of theimage of a patch of faces by the action of the dual map:

X c = lim M−n(πc(E∗2 (σ)n(U)))

the same property is true for the expanding Rauzy fractal.

Applications

The window

-1 -0.5 0.5 1

0.5 Πce1

Sc (2 ∧ 1)

-1 -0.5 0.5 1

0.5 Πce1

Sc (1 ∧ 3)

-1 -0.5 0.5 1

0.5 Πce1

Sc (4 ∧ 1)

-1 -0.5 0.5 1

0.5 Πce1

Sc (3 ∧ 2)

-1 -0.5 0.5 1

0.5 Πce1

Sc (2 ∧ 4)

-1 -0.5 0.5 1

0.5 Πce1

Sc (4 ∧ 3)

-1 -0.5 0.5 1

0.5 Πce1

⋃i∧j∈Vc

Sc (i ∧ j)

Figure 21: Sc (i ∧ j)

Applications

renormalization and projection

-1.5 -1 -0.5 0.5 1 1.5

U (1)c

-1.5 -1 -0.5 0.5 1 1.5

E2 (θ)U (2)c

-1.5 -1 -0.5 0.5 1 1.5

U (2)c

-1.5 -1 -0.5 0.5 1 1.5

E2 (θ)U (3)c

-1.5 -1 -0.5 0.5 1 1.5

U (3)c

-1.5 -1 -0.5 0.5 1 1.5

E2 (θ)U (4)c

-1.5 -1 -0.5 0.5 1 1.5

U (4)c

-1.5 -1 -0.5 0.5 1 1.5

E2 (θ)U (5)c

-1.5 -1 -0.5 0.5 1 1.5

U (6)c

-1.5 -1 -0.5 0.5 1 1.5

E2 (θ)U (6)c

-5 -4-3.5-3 2.5-2-1.5-1-0.50.511.522.533.54 5

-2-1.5

-1-0.5

E29 (θ)U (1)

-5 -4-3.5-3 2.5-2-1.5-1-0.50.511.522.533.54 5

-2-1.5

-1-0.5

E28 (θ)U (8)

-4.5-4-3.5-3 2.5-2-1.5-1-0.5 0.511.522.533.5

-3.5-3

-2.5-2

-1.5-1

E27 (θ)U (7)

-4 -3 -2 -1 1 2 3 4

E210 (θ)U (2)

Applications

-7 -6 -5 -4 -3 -2 -1 1 2 3 4 5

E211 (θ)U (3)

-7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8

E212 (θ)U (4)

-8 7 8

E213 (θ)U (5)

Figure 24:

-1 -0.5 0.5 1x

Πce4! -1 -0.5 0.5 1

-1 -0.5 0.5 1x

Figure 25: 2 hexagonal patches of U (6)c

Applications

-1 -0.5 0.5 1

Xc (2 ∧ 1)

-1 -0.5 0.5 1

Xc (1 ∧ 3)

-1 -0.5 0.5 1

Xc (4 ∧ 1)

-1 -0.5 0.5 1

Xc (3 ∧ 2)

-1 -0.5 0.5 1

Xc (2 ∧ 4)

-1 -0.5 0.5 1

Xc (4 ∧ 3)

-1 -0.5 0.5 1

⋃i∧j∈Vc

Xc (i ∧ 1)

Figure 10: Xc (i ∧ j)

Applications

The other window

-1 -0.5 0.5

0.5Πee1

Se (ϕ (4 ∧ 3))

-1 -0.5 0.5

0.5Πee1

Se (ϕ (2 ∧ 4))

-1 -0.5 0.5

0.5Πee1

Se (ϕ (3 ∧ 2))

-1 -0.5 0.5

0.5Πee1

Se (ϕ (4 ∧ 1))

-1 -0.5 0.5

0.5Πee1

Se (ϕ (1 ∧ 3))

-1 -0.5 0.5

0.5Πee1

Se (ϕ (2 ∧ 1))

-1 -0.5 0.5

0.5Πee1

⋃i∧j∈Ve

Se (ϕ (i ∧ j))

Figure 20: Se (ϕ (i ∧ j))

Applications

Arithmetics: complex β-expansion

There is an associated complex β-expansion, whose domain is thegeneralized Rauzy fractal.In this expansion, any complex number can be written in a(almost) unique way:

∞∑n=N0

εnµn

where µ is the small complex eigenvalue of the matrix M, and εnbelongs to a finite set {0, f1, f2} and satisfies a Markov condition,related to the GIFS.

Applications

Symbolic dynamics

By taking the product of the corresponding Rauzy fractals:

X ci∧j × X e

one obtains a partition of the torus T4.This partition gives a symbolic dynamics for the action of thematrix A which is a subshift of finite type.This is the first known explicit Markov partition for a non-Pisotirreducible automorphism of the torus.It is the natural extension of the β-expansion.

Applications

Transversal dynamics

I Study the transversal flow of these tilings.

I Find a good symbolic dynamics for this R2-action

I Meaning unclear:

I pseudo-group of translations?

Applications

I Meaning unclear:

Applications

I Meaning unclear:

Applications

I Meaning unclear:

Applications

-1.5 -1 -0.5 0.5 1 1.5

U (1)c

-1.5 -1 -0.5 0.5 1 1.5

E2 (θ)U (2)c

-1.5 -1 -0.5 0.5 1 1.5

U (2)c

-1.5 -1 -0.5 0.5 1 1.5

E2 (θ)U (3)c

-1.5 -1 -0.5 0.5 1 1.5

U (3)c

-1.5 -1 -0.5 0.5 1 1.5

E2 (θ)U (4)c

-1.5 -1 -0.5 0.5 1 1.5

U (4)c

-1.5 -1 -0.5 0.5 1 1.5

E2 (θ)U (5)c

-1.5 -1 -0.5 0.5 1 1.5

U (6)c

-1.5 -1 -0.5 0.5 1 1.5

E2 (θ)U (6)c

-5 -4-3.5-3 2.5-2-1.5-1-0.50.511.522.533.54 5

-2-1.5

-1-0.5

E29 (θ)U (1)

-5 -4-3.5-3 2.5-2-1.5-1-0.50.511.522.533.54 5

-2-1.5

-1-0.5

E28 (θ)U (8)

-4.5-4-3.5-3 2.5-2-1.5-1-0.5 0.511.522.533.5

-3.5-3

-2.5-2

-1.5-1

E27 (θ)U (7)

-4 -3 -2 -1 1 2 3 4

E210 (θ)U (2)

Applications

Classification problems

I Which automorphisms have positive E2(σ)?

I Do all of them define a fractal tiling?

I What of those with positive matrix A: can we get rid ofcancellations?

I In the general case, can we obtain results by grouping tiles(blocking)?

Applications

Arithmetic problems

I Can we get completely real examples?

I What can be said on the number system?

I Can we obtain good approximation in this way?

I Can we go beyond the algebraic case (generalized continuedfractions)?

Applications

Arithmetic problems

Applications

Arithmetic problems

Applications

Arithmetic problems

Applications

Word problems

I Can we define a good structure of 2-dimensional wordscorresponding to the tilings (underlying lattice)?

I Can we define a reasonable notion of complexity?

I Can we define discrete dynamical systems (transversaldynamics)?

I Can we find an equivalent of sturmian sequences in thissetting?

Applications

Word problems

Applications

Word problems

Applications

Word problems

general theory of unitary pisot substitution · applications low complexity and sturmian sequences...

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