The Structure of Hyperbolic Attractors on Surfaces

The structure of hyperbolic attractors on surfaces

Department of Mathematics University of Maryland, College Park

The structure of hyperbolic attractors on surfaces – p. 1/21 Examples Previous Results Outline of Argument and open questions

Outline of Talk

Background

The structure of hyperbolic attractors on surfaces – p. 2/21 Previous Results Outline of Argument and open questions

Outline of Talk

Background Examples

The structure of hyperbolic attractors on surfaces – p. 2/21 Outline of Argument and open questions

Outline of Talk

Background Examples Previous Results

The structure of hyperbolic attractors on surfaces – p. 2/21 Outline of Talk

Background Examples Previous Results Outline of Argument and open questions

The structure of hyperbolic attractors on surfaces – p. 2/21 Standing Assumptions

M is a compact smooth boundaryless surface. f ∈ Di(M)

The structure of hyperbolic attractors on surfaces – p. 3/21 If M is a compact manifold and f a diffeomorphism of M, then equivalently we say that given two open sets U and V in X ∃ n ∈ N such that f n(U) ∩ V =6 ∅. Deﬁnition: A set X is mixing (topologically) if for any open sets U and V in X ∃ N ∈ N such that ∀ n N f n(U) ∩ V =6 ∅.

Transitivity and Mixing

Deﬁnition: A set X is transitive for a map f if there is a point x ∈ X with a dense forward orbit

The structure of hyperbolic attractors on surfaces – p. 4/21 Deﬁnition: A set X is mixing (topologically) if for any open sets U and V in X ∃ N ∈ N such that ∀ n N f n(U) ∩ V =6 ∅.

Transitivity and Mixing

The structure of hyperbolic attractors on surfaces – p. 4/21 Transitivity and Mixing

Deﬁnition: A set X is transitive for a map f if there is a point x ∈ X with a dense forward orbit If M is a compact manifold and f a diffeomorphism of M, then equivalently we say that given two open sets U and V in X ∃ n ∈ N such that f n(U) ∩ V =6 ∅. Deﬁnition: A set X is mixing (topologically) if for any open sets U and V in X ∃ N ∈ N such that ∀ n N f n(U) ∩ V =6 ∅.

The structure of hyperbolic attractors on surfaces – p. 4/21 Deﬁnition: A set X is an attractor for a map f if ∃ neighborhood U (an attracting set) of X such n that X = n∈N f (U) and f(U) U. Deﬁnition: A set is a hyperbolic attractor if is transitive Tand has an attracting set U.

Hyperbolic Attractors

The structure of hyperbolic attractors on surfaces – p. 5/21 Deﬁnition: A set is a hyperbolic attractor if is transitive and has an attracting set U.

Hyperbolic Attractors

Deﬁnition: A compact invariant set for f ∈ Di(M) is hyperbolic if the tangent space has a continuous invariant splitting s u s TM = E E where E is uniformly contracting and Eu is uniformly expanding. Deﬁnition: A set X is an attractor for a map f if ∃ neighborhood U (an attracting set) of X such n that X = n∈N f (U) and f(U) U. T

The structure of hyperbolic attractors on surfaces – p. 5/21 Hyperbolic Attractors

The structure of hyperbolic attractors on surfaces – p. 5/21 If is mixing we know W u(x) = and W s(x) is dense in W s() ∀ x ∈ . k = i=1 i where i’s are compact disjoint, k f(i) = i+1, and f (i) is a mixing hyperbolic attractorS .

Properties of Hyperbolic Attractors

For each x ∈ we know W u(x)

The structure of hyperbolic attractors on surfaces – p. 6/21 k = i=1 i where i’s are compact disjoint, k f(i) = i+1, and f (i) is a mixing hyperbolic attractorS .

Properties of Hyperbolic Attractors

For each x ∈ we know W u(x) If is mixing we know W u(x) = and W s(x) is dense in W s() ∀ x ∈ .

The structure of hyperbolic attractors on surfaces – p. 6/21 Properties of Hyperbolic Attractors

For each x ∈ we know W u(x) If is mixing we know W u(x) = and W s(x) is dense in W s() ∀ x ∈ . k = i=1 i where i’s are compact disjoint, k f(i) = i+1, and f (i) is a mixing hyperbolic attractorS .

The structure of hyperbolic attractors on surfaces – p. 6/21 int(Ri) ∩ int(Rj) = ∅ if i =6 j for some sufﬁciently small Ri is u s (W (x) ∩ Ri) (W (x) ∩ Ri) x ∈ Ri, f(x) ∈ Rj, and i → j is an allowed transition, then s 1 u f(W (x, Ri)) Rj and f (W (f(x), Rj)) Ri.

Markov Partition

Decomposition of into ﬁnite number of dynamical rectangles R1, ..., Rn such that for each 1 i n

The structure of hyperbolic attractors on surfaces – p. 7/21 for some sufﬁciently small Ri is u s (W (x) ∩ Ri) (W (x) ∩ Ri) x ∈ Ri, f(x) ∈ Rj, and i → j is an allowed transition, then s 1 u f(W (x, Ri)) Rj and f (W (f(x), Rj)) Ri.

Markov Partition

Decomposition of into ﬁnite number of dynamical rectangles R1, ..., Rn such that for each 1 i n int(Ri) ∩ int(Rj) = ∅ if i =6 j

The structure of hyperbolic attractors on surfaces – p. 7/21 x ∈ Ri, f(x) ∈ Rj, and i → j is an allowed transition, then s 1 u f(W (x, Ri)) Rj and f (W (f(x), Rj)) Ri.

Markov Partition

Decomposition of into ﬁnite number of dynamical rectangles R1, ..., Rn such that for each 1 i n int(Ri) ∩ int(Rj) = ∅ if i =6 j

for some sufﬁciently small Ri is u s (W (x) ∩ Ri) (W (x) ∩ Ri)

The structure of hyperbolic attractors on surfaces – p. 7/21 Markov Partition

Decomposition of into ﬁnite number of dynamical rectangles R1, ..., Rn such that for each 1 i n int(Ri) ∩ int(Rj) = ∅ if i =6 j for some sufﬁciently small Ri is u s (W (x) ∩ Ri) (W (x) ∩ Ri) x ∈ Ri, f(x) ∈ Rj, and i → j is an allowed transition, then

s 1 u f(W (x, Ri)) Rj and f (W (f(x), Rj)) Ri.

The structure of hyperbolic attractors on surfaces – p. 7/21 In general these address the question where we assume we have an attractor, then what can be said about the set and when are two sets homeomorphic. Question: Suppose we know the topology of and is hyperbolic, what can be concluded about the set?

General Question

The structure of hyperbolic attractors on surfaces has been studied extensively by Plykin, Bonatti, Williams, Zhirov, Grines, F. and J. Rodriquez-Hertz, and others.

The structure of hyperbolic attractors on surfaces – p. 8/21 Question: Suppose we know the topology of and is hyperbolic, what can be concluded about the set?

General Question

The structure of hyperbolic attractors on surfaces has been studied extensively by Plykin, Bonatti, Williams, Zhirov, Grines, F. and J. Rodriquez-Hertz, and others. In general these address the question where we assume we have an attractor, then what can be said about the set and when are two sets homeomorphic.

The structure of hyperbolic attractors on surfaces – p. 8/21 General Question

The structure of hyperbolic attractors on surfaces – p. 8/21 A nontrivial attractor means not the orbit of a periodic sink Counterexamples in higher dimensions

Statement of Main Result

Theorem 1:(F.) If M is a compact surface and is a nontrivial mixing hyperbolic attractor for a diffeomorphism f of M, and is hyperbolic for a diffeomorphism g of M, then is either a nontrivial mixing hyperbolic attractor or a nontrivial mixing hyperbolic repeller for g.

The structure of hyperbolic attractors on surfaces – p. 9/21 Counterexamples in higher dimensions

Statement of Main Result

The structure of hyperbolic attractors on surfaces – p. 9/21 Statement of Main Result

The structure of hyperbolic attractors on surfaces – p. 9/21 Non-mixing Case

Theorem 2:(F.) If M is a compact surface and is a nontrivial hyperbolic attractor for a diffeomorphism f of M, and is hyperbolic for a diffeomorphism g of M, then there exists an N N n ∈ and sets 1, ..., N where = i=1 i, i ∩ j = ∅ if i =6 j, and each i is a mixing hyperbolic attractor or repeller for gn.S

The structure of hyperbolic attractors on surfaces – p. 10/21 Commuting Diffeo. r 2

Theorem 3:(F.) Let M be a compact surface and is a nontrivial hyperbolic attractor for f ∈ Dir(M), r 2. Then there exists a neighborhood U of f in Dir(M) and an open and dense set U 0 U such that for all f 0 ∈ U if g ∈ Di1(M) where f 0g = gf 0 (g in the centralizer 0 j of f), then g|W s() = (f ) for some j ∈ Z. This is an extension of a result from the work of Palis and Yoccoz.

The structure of hyperbolic attractors on surfaces – p. 11/21 p 1

V p p 2 n T2 Let = n∈N f ( V ), then is a hyperbolic attractor. T

DA example

Let f : T2 → T2 Anosov map from map 2 1 A = , p a hyperbolic ﬁxed point of f. "1 1#

The structure of hyperbolic attractors on surfaces – p. 12/21 p 1

V p p 2 n T2 Let = n∈N f ( V ), then is a hyperbolic attractor. T

DA example

Let f : T2 → T2 Anosov map from map 2 1 A = , p a hyperbolic ﬁxed point of f. "1 1#

The structure of hyperbolic attractors on surfaces – p. 12/21 n T2 Let = n∈N f ( V ), then is a hyperbolic attractor. T

DA example

Let f : T2 → T2 Anosov map from map 2 1 A = , p a hyperbolic ﬁxed point of f. "1 1#

p 1

V p

p 2

The structure of hyperbolic attractors on surfaces – p. 12/21 DA example

Let f : T2 → T2 Anosov map from map 2 1 A = , p a hyperbolic ﬁxed point of f. "1 1#

p 1

V p

p 2 n T2 Let = n∈N f ( V ), then is a hyperbolic attractor. T

The structure of hyperbolic attractors on surfaces – p. 12/21 Plykin Attractor

Another example due to Plykin can be built on the 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 on the 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 on the 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 Plykin: Shows trapping region is homeomorphic to disk with holes, then must have at least 3 holes.

Results of Williams and Plykin

Williams: Shows that if x is in an attractor on a surface, then a neighborhood of x in the attractor is [0, 1] C for a Cantor set C. Also, uses symbolic dynamics to classify when two diffeomorphisms restricted to two attractors are conjugate.

The structure of hyperbolic attractors on surfaces – p. 14/21 Results of Williams and Plykin

The structure of hyperbolic attractors on surfaces – p. 14/21 F. and J. Rodriquez-Hertz: Obtain a local dynamical and topological picture of expansive attractors on surfaces.

Zhirov and F. and J. Rodriquez-Hertz

Zhirov: Classiﬁes when 2 diffeomorphisms are conjugate in neighborhood of attractor. Gives algorithm for the classiﬁcation.

The structure of hyperbolic attractors on surfaces – p. 15/21 Zhirov and F. and J. Rodriquez-Hertz

Zhirov: Classiﬁes when 2 diffeomorphisms are conjugate in neighborhood of attractor. Gives algorithm for the classiﬁcation. F. and J. Rodriquez-Hertz: Obtain a local dynamical and topological picture of expansive attractors on surfaces.

The structure of hyperbolic attractors on surfaces – p. 15/21 W u(x, g) =6 W u(x, f) or W s(x, g) =6 W u(x, f) for some x ∈

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 ∈

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 ∈ W u(x, g) =6 W u(x, f) or W s(x, g) =6 W u(x, f) for some x ∈

The structure of hyperbolic attractors on surfaces – p. 16/21 Since cl(W u(x, f)) = then can show mixing hyperbolic attractor.

Case 1

Assume W u(x, f) = W u(x, g) We show is attractor: Take > 0 sufﬁciently small and s V = x∈ W (x) is attracting set. i.e. - = gn(V ). Tn0 T

The structure of hyperbolic attractors on surfaces – p. 17/21 Case 1

Assume W u(x, f) = W u(x, g) We show is attractor: Take > 0 sufﬁciently small and s V = x∈ W (x) is attracting set. i.e. - = gn(V ). Tn0 SinceTcl(W u(x, f)) = then can show mixing hyperbolic attractor.

The structure of hyperbolic attractors on surfaces – p. 17/21 ˜ s u For x, y ∈ in same rectangle W (x) ∩ W (y) is one point in ˜. So product structure for points in same rectangle.

Markov Partition Theorem

Theorem 4:(F.) If is a hyperbolic set and V is a neighborhood of , then there exists a hyperbolic set ˜ with a Markov partition such that ˜ V .

The structure of hyperbolic attractors on surfaces – p. 18/21 Markov Partition Theorem

Theorem 4:(F.) If is a hyperbolic set and V is a neighborhood of , then there exists a hyperbolic set ˜ with a Markov partition such that ˜ V . ˜ s u For x, y ∈ in same rectangle W (x) ∩ W (y) is one point in ˜. So product structure for points in same rectangle.

The structure of hyperbolic attractors on surfaces – p. 18/21 Case 2

Take U neighborhood of and ˜ in U containing with Markov partition. Show there exists x ∈ such that x in interior of rectangle and W 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 2