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The structure ofhyperbolic attractors on
surfacesTodd Fisher
tfisher@math.umd.edu
Department of Mathematics
University of Maryland, College Park
The structure of hyperbolic attractors on surfaces – p. 1/21
Outline of TalkBackground
Examples
Previous Results
Outline of Argument and open questions
The structure of hyperbolic attractors on surfaces – p. 2/21
Outline of TalkBackground
Examples
Previous Results
Outline of Argument and open questions
The structure of hyperbolic attractors on surfaces – p. 2/21
Outline of TalkBackground
Examples
Previous Results
Outline of Argument and open questions
The structure of hyperbolic attractors on surfaces – p. 2/21
Outline of TalkBackground
Examples
Previous Results
Outline of Argument and open questions
The structure of hyperbolic attractors on surfaces – p. 2/21
StandingAssumptions
M is a compact smooth boundaryless surface.
f ∈ Diff(M)
The structure of hyperbolic attractors on surfaces – p. 3/21
Transitivity andMixing
Definition: A set X is transitive for a map f if thereis a point x ∈ X with a dense forward orbit
If M is a compact manifold and f adiffeomorphism of M , then equivalently we saythat given two open sets U and V in X ∃ n ∈ N
such that fn(U) ∩ V 6= ∅.
Definition: A set X is mixing (topologically) if forany open sets U and V in X ∃ N ∈ N such that ∀n ≥ N fn(U) ∩ V 6= ∅.
The structure of hyperbolic attractors on surfaces – p. 4/21
Transitivity andMixing
Definition: A set X is transitive for a map f if thereis a point x ∈ X with a dense forward orbit
If M is a compact manifold and f adiffeomorphism of M , then equivalently we saythat given two open sets U and V in X ∃ n ∈ N
such that fn(U) ∩ V 6= ∅.
Definition: A set X is mixing (topologically) if forany open sets U and V in X ∃ N ∈ N such that ∀n ≥ N fn(U) ∩ V 6= ∅.
The structure of hyperbolic attractors on surfaces – p. 4/21
Transitivity andMixing
Definition: A set X is transitive for a map f if thereis a point x ∈ X with a dense forward orbit
If M is a compact manifold and f adiffeomorphism of M , then equivalently we saythat given two open sets U and V in X ∃ n ∈ N
such that fn(U) ∩ V 6= ∅.
Definition: A set X is mixing (topologically) if forany open sets U and V in X ∃ N ∈ N such that ∀n ≥ N fn(U) ∩ V 6= ∅.
The structure of hyperbolic attractors on surfaces – p. 4/21
Hyperbolic Attractors
Definition: A compact invariant set Λ forf ∈ Diff(M) is hyperbolic if the tangent spacehas a continuous invariant splittingTΛM = E
s ⊕ Eu where E
s is uniformly contractingand E
u is uniformly expanding.
Definition: A set X is an attractor for a map f if ∃neighborhood U (an attracting set) of X suchthat X =
⋂
n∈Nfn(U) and f(U) ⊂ U .
Definition: A set Λ is a hyperbolic attractor if Λ istransitive and has an attracting set U .
The structure of hyperbolic attractors on surfaces – p. 5/21
Hyperbolic Attractors
Definition: A compact invariant set Λ forf ∈ Diff(M) is hyperbolic if the tangent spacehas a continuous invariant splittingTΛM = E
s ⊕ Eu where E
s is uniformly contractingand E
u is uniformly expanding.
Definition: A set X is an attractor for a map f if ∃neighborhood U (an attracting set) of X suchthat X =
⋂
n∈Nfn(U) and f(U) ⊂ U .
Definition: A set Λ is a hyperbolic attractor if Λ istransitive and has an attracting set U .
The structure of hyperbolic attractors on surfaces – p. 5/21
Hyperbolic Attractors
Definition: A compact invariant set Λ forf ∈ Diff(M) is hyperbolic if the tangent spacehas a continuous invariant splittingTΛM = E
s ⊕ Eu where E
s is uniformly contractingand E
u is uniformly expanding.
Definition: A set X is an attractor for a map f if ∃neighborhood U (an attracting set) of X suchthat X =
⋂
n∈Nfn(U) and f(U) ⊂ U .
Definition: A set Λ is a hyperbolic attractor if Λ istransitive and has an attracting set U .
The structure of hyperbolic attractors on surfaces – p. 5/21
Properties ofHyperbolic Attractors
For each x ∈ Λ we know W u(x) ⊂ Λ
If Λ is mixing we know W u(x) = Λ and W s(x) isdense in W s(Λ) ∀ x ∈ Λ.
Λ =⋃k
i=1 Λi where Λi’s are compact disjoint,f(Λi) = Λi+1, and f k(Λi) is a mixing hyperbolicattractor.
The structure of hyperbolic attractors on surfaces – p. 6/21
Properties ofHyperbolic Attractors
For each x ∈ Λ we know W u(x) ⊂ Λ
If Λ is mixing we know W u(x) = Λ and W s(x) isdense in W s(Λ) ∀ x ∈ Λ.
Λ =⋃k
i=1 Λi where Λi’s are compact disjoint,f(Λi) = Λi+1, and f k(Λi) is a mixing hyperbolicattractor.
The structure of hyperbolic attractors on surfaces – p. 6/21
Properties ofHyperbolic Attractors
For each x ∈ Λ we know W u(x) ⊂ Λ
If Λ is mixing we know W u(x) = Λ and W s(x) isdense in W s(Λ) ∀ x ∈ Λ.
Λ =⋃k
i=1 Λi where Λi’s are compact disjoint,f(Λi) = Λi+1, and f k(Λi) is a mixing hyperbolicattractor.
The structure of hyperbolic attractors on surfaces – p. 6/21
Markov PartitionDecomposition of Λ into finite number ofdynamical rectangles R1, ..., Rn such that foreach 1 ≤ i ≤ n
int(Ri) ∩ int(Rj) = ∅ if i 6= j
for some ε sufficiently small Ri is(W u
ε (x) ∩ Ri) × (W sε (x) ∩ Ri)
x ∈ Ri, f(x) ∈ Rj, and i → j is an allowedtransition, then
f(W s(x, Ri)) ⊂ Rj and f−1(W u(f(x), Rj)) ⊂ Ri.
The structure of hyperbolic attractors on surfaces – p. 7/21
Markov PartitionDecomposition of Λ into finite number ofdynamical rectangles R1, ..., Rn such that foreach 1 ≤ i ≤ nint(Ri) ∩ int(Rj) = ∅ if i 6= j
for some ε sufficiently small Ri is(W u
ε (x) ∩ Ri) × (W sε (x) ∩ Ri)
x ∈ Ri, f(x) ∈ Rj, and i → j is an allowedtransition, then
f(W s(x, Ri)) ⊂ Rj and f−1(W u(f(x), Rj)) ⊂ Ri.
The structure of hyperbolic attractors on surfaces – p. 7/21
Markov PartitionDecomposition of Λ into finite number ofdynamical rectangles R1, ..., Rn such that foreach 1 ≤ i ≤ nint(Ri) ∩ int(Rj) = ∅ if i 6= j
for some ε sufficiently small Ri is(W u
ε (x) ∩ Ri) × (W sε (x) ∩ Ri)
x ∈ Ri, f(x) ∈ Rj, and i → j is an allowedtransition, then
f(W s(x, Ri)) ⊂ Rj and f−1(W u(f(x), Rj)) ⊂ Ri.
The structure of hyperbolic attractors on surfaces – p. 7/21
Markov PartitionDecomposition of Λ into finite number ofdynamical rectangles R1, ..., Rn such that foreach 1 ≤ i ≤ nint(Ri) ∩ int(Rj) = ∅ if i 6= j
for some ε sufficiently small Ri is(W u
ε (x) ∩ Ri) × (W sε (x) ∩ Ri)
x ∈ Ri, f(x) ∈ Rj, and i → j is an allowedtransition, then
f(W s(x, Ri)) ⊂ Rj and f−1(W u(f(x), Rj)) ⊂ Ri.
The structure of hyperbolic attractors on surfaces – p. 7/21
General QuestionThe structure of hyperbolic attractors on surfaceshas been studied extensively by Plykin, Bonatti,Williams, Zhirov, Grines, F. and J.Rodriquez-Hertz, and others.
In general these address the question where weassume we have an attractor, then what can besaid about the set and when are two setshomeomorphic.
Question: Suppose we know the topology of Λand is hyperbolic, what can be concluded aboutthe set?
The structure of hyperbolic attractors on surfaces – p. 8/21
General QuestionThe structure of hyperbolic attractors on surfaceshas been studied extensively by Plykin, Bonatti,Williams, Zhirov, Grines, F. and J.Rodriquez-Hertz, and others.
In general these address the question where weassume we have an attractor, then what can besaid about the set and when are two setshomeomorphic.
Question: Suppose we know the topology of Λand is hyperbolic, what can be concluded aboutthe set?
The structure of hyperbolic attractors on surfaces – p. 8/21
General QuestionThe structure of hyperbolic attractors on surfaceshas been studied extensively by Plykin, Bonatti,Williams, Zhirov, Grines, F. and J.Rodriquez-Hertz, and others.
In general these address the question where weassume we have an attractor, then what can besaid about the set and when are two setshomeomorphic.
Question: Suppose we know the topology of Λand is hyperbolic, what can be concluded aboutthe set?
The structure of hyperbolic attractors on surfaces – p. 8/21
Statement of MainResult
Theorem 1:(F.) If M is a compact surface and Λ isa nontrivial mixing hyperbolic attractor for adiffeomorphism f of M , and Λ is hyperbolic for adiffeomorphism g of M , then Λ is either anontrivial mixing hyperbolic attractor or anontrivial mixing hyperbolic repeller for g.
A nontrivial attractor means not the orbit of aperiodic sinkCounterexamples in higher dimensions
The structure of hyperbolic attractors on surfaces – p. 9/21
Statement of MainResult
Theorem 1:(F.) If M is a compact surface and Λ isa nontrivial mixing hyperbolic attractor for adiffeomorphism f of M , and Λ is hyperbolic for adiffeomorphism g of M , then Λ is either anontrivial mixing hyperbolic attractor or anontrivial mixing hyperbolic repeller for g.
A nontrivial attractor means not the orbit of aperiodic sink
Counterexamples in higher dimensions
The structure of hyperbolic attractors on surfaces – p. 9/21
Statement of MainResult
Theorem 1:(F.) If M is a compact surface and Λ isa nontrivial mixing hyperbolic attractor for adiffeomorphism f of M , and Λ is hyperbolic for adiffeomorphism g of M , then Λ is either anontrivial mixing hyperbolic attractor or anontrivial mixing hyperbolic repeller for g.
A nontrivial attractor means not the orbit of aperiodic sinkCounterexamples in higher dimensions
The structure of hyperbolic attractors on surfaces – p. 9/21
Non-mixing Case
Theorem 2:(F.) If M is a compact surface and Λ isa nontrivial hyperbolic attractor for adiffeomorphism f of M , and Λ is hyperbolic for adiffeomorphism g of M , then there exists ann ∈ N and sets Λ1, ..., ΛN where Λ =
⋃Ni=1 Λi,
Λi ∩ Λj = ∅ if i 6= j, and each Λi is a mixinghyperbolic attractor or repeller for gn.
The structure of hyperbolic attractors on surfaces – p. 10/21
Commuting Diffeo.r ≥ 2
Theorem 3:(F.) Let M be a compact surface and Λis a nontrivial hyperbolic attractor forf ∈ Diffr(M), r ≥ 2. Then there exists aneighborhood U of f in Diffr(M) and an openand dense set U ′ ⊂ U such that for all f ′ ∈ U ifg ∈ Diff1(M) where f ′g = gf ′ (g in the centralizerof f ), then g|W s(Λ) = (f ′)j for some j ∈ Z.
This is an extension of a result from the work ofPalis and Yoccoz.
The structure of hyperbolic attractors on surfaces – p. 11/21
DA example
Let f : T2 → T
2 Anosov map from map
A =
[
2 1
1 1
]
, p a hyperbolic fixed point of f .
pV
p
p
1
2
Let Λ =⋂
n∈Nfn(T2 − V ), then Λ is a hyperbolic
attractor.
The structure of hyperbolic attractors on surfaces – p. 12/21
DA example
Let f : T2 → T
2 Anosov map from map
A =
[
2 1
1 1
]
, p a hyperbolic fixed point of f .
p
pV
p
p
1
2
Let Λ =⋂
n∈Nfn(T2 − V ), then Λ is a hyperbolic
attractor.
The structure of hyperbolic attractors on surfaces – p. 12/21
DA example
Let f : T2 → T
2 Anosov map from map
A =
[
2 1
1 1
]
, p a hyperbolic fixed point of f .
pV
p
p
1
2
Let Λ =⋂
n∈Nfn(T2 − V ), then Λ is a hyperbolic
attractor.
The structure of hyperbolic attractors on surfaces – p. 12/21
DA example
Let f : T2 → T
2 Anosov map from map
A =
[
2 1
1 1
]
, p a hyperbolic fixed point of f .
pV
p
p
1
2
Let Λ =⋂
n∈Nfn(T2 − V ), then Λ is a hyperbolic
attractor.
The structure of hyperbolic attractors on surfaces – p. 12/21
Plykin Attractor
Another example due to Plykin can be built onthe disk, so embedded into any surface. Note:we need three holes in our domain.
The structure of hyperbolic attractors on surfaces – p. 13/21
Plykin Attractor
Another example due to Plykin can be built onthe disk, so embedded into any surface. Note:we need three holes in our domain.
V
The structure of hyperbolic attractors on surfaces – p. 13/21
Plykin Attractor
Another example due to Plykin can be built onthe disk, so embedded into any surface. Note:we need three holes in our domain.
V
f(V)
The structure of hyperbolic attractors on surfaces – p. 13/21
Results of Williamsand Plykin
Williams: Shows that if x is in an attractor on asurface, then a neighborhood of x in the attractoris [0, 1] × C for a Cantor set C. Also, usessymbolic dynamics to classify when twodiffeomorphisms restricted to two attractors areconjugate.
Plykin: Shows trapping region is homeomorphicto disk with holes, then must have at least 3holes.
The structure of hyperbolic attractors on surfaces – p. 14/21
Results of Williamsand Plykin
Williams: Shows that if x is in an attractor on asurface, then a neighborhood of x in the attractoris [0, 1] × C for a Cantor set C. Also, usessymbolic dynamics to classify when twodiffeomorphisms restricted to two attractors areconjugate.
Plykin: Shows trapping region is homeomorphicto disk with holes, then must have at least 3holes.
The structure of hyperbolic attractors on surfaces – p. 14/21
Zhirov and F. and J.Rodriquez-Hertz
Zhirov: Classifies when 2 diffeomorphisms areconjugate in neighborhood of attractor. Givesalgorithm for the classification.
F. and J. Rodriquez-Hertz: Obtain a local dynamicaland topological picture of expansive attractors onsurfaces.
The structure of hyperbolic attractors on surfaces – p. 15/21
Zhirov and F. and J.Rodriquez-Hertz
Zhirov: Classifies when 2 diffeomorphisms areconjugate in neighborhood of attractor. Givesalgorithm for the classification.F. and J. Rodriquez-Hertz: Obtain a local dynamicaland topological picture of expansive attractors onsurfaces.
The structure of hyperbolic attractors on surfaces – p. 15/21
Outline of argument -2 cases
W u(x, g) = W u(x, f) or W s(x, g) = W u(x, f)for all x ∈ Λ
W u(x, g) 6= W u(x, f) or W s(x, g) 6= W u(x, f)for some x ∈ Λ
The structure of hyperbolic attractors on surfaces – p. 16/21
Outline of argument -2 cases
W u(x, g) = W u(x, f) or W s(x, g) = W u(x, f)for all x ∈ Λ
W u(x, g) 6= W u(x, f) or W s(x, g) 6= W u(x, f)for some x ∈ Λ
The structure of hyperbolic attractors on surfaces – p. 16/21
Case 1Assume W u(x, f) = W u(x, g) We show Λ isattractor: Take ε > 0 sufficiently small andV =
⋂
x∈Λ W sε (x) is attracting set. i.e. -
Λ =⋂
n≥0 gn(V ).
Since cl(W u(x, f)) = Λ then can show Λ mixinghyperbolic attractor.
The structure of hyperbolic attractors on surfaces – p. 17/21
Case 1Assume W u(x, f) = W u(x, g) We show Λ isattractor: Take ε > 0 sufficiently small andV =
⋂
x∈Λ W sε (x) is attracting set. i.e. -
Λ =⋂
n≥0 gn(V ).
Since cl(W u(x, f)) = Λ then can show Λ mixinghyperbolic attractor.
The structure of hyperbolic attractors on surfaces – p. 17/21
Markov PartitionTheorem
Theorem 4:(F.) If Λ is a hyperbolic set and V is aneighborhood of Λ, then there exists a hyperbolicset Λ̃ with a Markov partition such thatΛ ⊂ Λ̃ ⊂ V .
For x, y ∈ Λ̃ in same rectangle W sε (x) ∩ W u
ε (y) isone point in Λ̃. So product structure for points insame rectangle.
The structure of hyperbolic attractors on surfaces – p. 18/21
Markov PartitionTheorem
Theorem 4:(F.) If Λ is a hyperbolic set and V is aneighborhood of Λ, then there exists a hyperbolicset Λ̃ with a Markov partition such thatΛ ⊂ Λ̃ ⊂ V .
For x, y ∈ Λ̃ in same rectangle W sε (x) ∩ W u
ε (y) isone point in Λ̃. So product structure for points insame rectangle.
The structure of hyperbolic attractors on surfaces – p. 18/21
Case 2Take U neighborhood of Λ and Λ̃ in U containingΛ with Markov partition. Show there exists x ∈ Λsuch that x in interior of rectangle andW u(x, g) 6= W u(x, f) or W s(x, g) 6= W u(x, f).
The structure of hyperbolic attractors on surfaces – p. 19/21
Case 2Take U neighborhood of Λ and Λ̃ in U containingΛ with Markov partition. Show there exists x ∈ Λsuch that x in interior of rectangle andW u(x, g) 6= W u(x, f) or W s(x, g) 6= W u(x, f).
x
s
u
uW (x,f)
W (x,g)
W (x,g)
The structure of hyperbolic attractors on surfaces – p. 19/21
Case 2Take U neighborhood of Λ and Λ̃ in U containingΛ with Markov partition. Show there exists x ∈ Λsuch that x in interior of rectangle andW u(x, g) 6= W u(x, f) or W s(x, g) 6= W u(x, f).
y
x
s
u
uW (x,f)
W (x,g)
W (x,g)
The structure of hyperbolic attractors on surfaces – p. 19/21
Case 2Take U neighborhood of Λ and Λ̃ in U containingΛ with Markov partition. Show there exists x ∈ Λsuch that x in interior of rectangle andW u(x, g) 6= W u(x, f) or W s(x, g) 6= W u(x, f).
x’
y
x
s
u
uW (x,f)
W (x,g)
W (x,g)
The structure of hyperbolic attractors on surfaces – p. 19/21
Case 2Take U neighborhood of Λ and Λ̃ in U containingΛ with Markov partition. Show there exists x ∈ Λsuch that x in interior of rectangle andW u(x, g) 6= W u(x, f) or W s(x, g) 6= W u(x, f).
x
s
u
uW (x,f)
W (x,g)
W (x,g)
The structure of hyperbolic attractors on surfaces – p. 19/21
Case 2Take U neighborhood of Λ and Λ̃ in U containingΛ with Markov partition. Show there exists x ∈ Λsuch that x in interior of rectangle andW u(x, g) 6= W u(x, f) or W s(x, g) 6= W u(x, f).
y
x
s
u
uW (x,f)
W (x,g)
W (x,g)
The structure of hyperbolic attractors on surfaces – p. 19/21
Case 2Take U neighborhood of Λ and Λ̃ in U containingΛ with Markov partition. Show there exists x ∈ Λsuch that x in interior of rectangle andW u(x, g) 6= W u(x, f) or W s(x, g) 6= W u(x, f).
x
s
u
uW (x,f)
W (x,g)
W (x,g)
The structure of hyperbolic attractors on surfaces – p. 19/21
Case 2Take U neighborhood of Λ and Λ̃ in U containingΛ with Markov partition. Show there exists x ∈ Λsuch that x in interior of rectangle andW u(x, g) 6= W u(x, f) or W s(x, g) 6= W u(x, f).
x
s
u
uW (x,f)
W (x,g)
W (x,g)
The structure of hyperbolic attractors on surfaces – p. 19/21
ContradictionSo Λ̃ has interior. Then contains hyperbolicattractor and repeller.
Since U arbitrarily small we show this implies Λhas attractor and repeller
Using density of W u(x, f) in Λ we seecontradiction.
The structure of hyperbolic attractors on surfaces – p. 20/21
ContradictionSo Λ̃ has interior. Then contains hyperbolicattractor and repeller.
Since U arbitrarily small we show this implies Λhas attractor and repeller
Using density of W u(x, f) in Λ we seecontradiction.
The structure of hyperbolic attractors on surfaces – p. 20/21
ContradictionSo Λ̃ has interior. Then contains hyperbolicattractor and repeller.
Since U arbitrarily small we show this implies Λhas attractor and repeller
Using density of W u(x, f) in Λ we seecontradiction.
The structure of hyperbolic attractors on surfaces – p. 20/21
Open Questions
Does the main Theorem hold for codimensionone attractors?
If a hyperbolic set is locally maximal (isolated)for f and hyperbolic for g is it necessarilylocally maximal for g?
The structure of hyperbolic attractors on surfaces – p. 21/21
Open Questions
Does the main Theorem hold for codimensionone attractors?
If a hyperbolic set is locally maximal (isolated)for f and hyperbolic for g is it necessarilylocally maximal for g?
The structure of hyperbolic attractors on surfaces – p. 21/21
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