The Structure of Hyperbolic Sets

The Structure of Hyperbolic Sets p. 1/35 The Structure of Hyperbolic Sets Todd Fisher tfisher@math.umd.edu Department of Mathematics University of Maryland, College Park

The Structure of Hyperbolic Sets p. 2/35 Outline of Talk History and Examples

The Structure of Hyperbolic Sets p. 2/35 Outline of Talk History and Examples Properties

The Structure of Hyperbolic Sets p. 2/35 Outline of Talk History and Examples Properties Structure

The Structure of Hyperbolic Sets p. 3/35 3 ways dynamics studied Measurable: Probabilistic and statistical properties.

The Structure of Hyperbolic Sets p. 3/35 3 ways dynamics studied Measurable: Probabilistic and statistical properties. Topological: This studies functions that are only assumed to be continuous.

The Structure of Hyperbolic Sets p. 3/35 3 ways dynamics studied Measurable: Probabilistic and statistical properties. Topological: This studies functions that are only assumed to be continuous. Smooth: Assume there is a derivative at every point.

The Structure of Hyperbolic Sets p. 4/35 Advantages to Smooth The local picture given by derivative

The Structure of Hyperbolic Sets p. 4/35 Advantages to Smooth The local picture given by derivative Very useful in hyperbolic case. Tangent space T Λ M splits into expanding E u and contracting directions E s.

The Structure of Hyperbolic Sets p. 5/35 Stable set and Unstable set The stable set of a point x M is W s (x) = {y M d(f n (x),f n (y)) 0 as n }.

The Structure of Hyperbolic Sets p. 5/35 Stable set and Unstable set The stable set of a point x M is W s (x) = {y M d(f n (x),f n (y)) 0 as n }. The unstable set of a point x M is W u (x) = {y M d(f n (x),f n (y)) 0 as n }.

The Structure of Hyperbolic Sets p. 6/35 One Origin - Celestial Mechanics In 19th century Poincaré began to study stability of solar system.

The Structure of Hyperbolic Sets p. 6/35 One Origin - Celestial Mechanics In 19th century Poincaré began to study stability of solar system. For a flow from a differential equation with fixed hyperbolic saddle point p and point x W s (p) W u (p).

The Structure of Hyperbolic Sets p. 7/35 Transverse Homoclinic Point If we look at a function f picture can become more complicated. This was in some sense the start of chaotic dynamics. A point x W s (p) W u (p) is called a transverse homoclinic point.

The Structure of Hyperbolic Sets p. 8/35 Homoclinic Tangle x p

The Structure of Hyperbolic Sets p. 8/35 Homoclinic Tangle x f(x) p

The Structure of Hyperbolic Sets p. 9/35 Further Results on Homoclinic Points In 1930 s Birkhoff showed that near a transverse homoclinic point p n x such that p n periodic

The Structure of Hyperbolic Sets p. 9/35 Further Results on Homoclinic Points In 1930 s Birkhoff showed that near a transverse homoclinic point p n x such that p n periodic In 1960 s Smale showed the following:

Smale s Horseshoe Smale generalized picture as follows: Take the unit square R = [0, 1] [0, 1] map the square as shown below. f(r) A 0000000 1111111 0000000 1111111 0000000 1111111 0000000 1111111 000000 111111 0000000 1111111 000000 111111 0000000 1111111 000000 111111 0000000 1111111 000000 111111 0000000 1111111 000000 111111 0000000 1111111 000000 111111 000000 111111 R 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 00000000 11111111 000000 111111 00000000 11111111 000000 111111 00000000 11111111 000000 111111 00000000 11111111 000000 111111 00000000 11111111 000000 111111 00000000 11111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 000000 111111 B There exist two region [ A and ] B in R such that 1/3 0 f A and f B looks like 0 3 The Structure of Hyperbolic Sets p. 10/35

The Structure of Hyperbolic Sets p. 11/35 Invariant Set for Horseshoe - 1 We want points that never leave R. Λ = n Zf n (R)

Invariant Set for Horseshoe -3 The Structure of Hyperbolic Sets p. 12/35

The Structure of Hyperbolic Sets p. 13/35 Dynamics of Horseshoe Then Λ is Middle Thirds Cantor Middle Thirds Cantor

The Structure of Hyperbolic Sets p. 13/35 Dynamics of Horseshoe Then Λ is Middle Thirds Cantor Middle Thirds Cantor The set Λ is chaotic in the sense of Devaney. periodic points of Λ are dense there is a point with a dense orbit (transitive)

The Structure of Hyperbolic Sets p. 14/35 Hyperbolic Set Hyperbolic A compact set Λ is hyperbolic if it is invariant (f(λ) = Λ) and the tangent space has a continuous invariant splitting T Λ M = E s E u where E s is uniformly contracting and E u is uniformly expanding. So C > 0 and λ (0, 1) such that: Dfxv n Cλ n v v E s x, n N and v Cλ n v v E u x, n N Df n x

The Structure of Hyperbolic Sets p. 15/35 Hyperbolic Properties -1 For a point of a hyperbolic set x Λ the stable and unstable sets are immersed copies of R m and R n where m = dim(e s ) and n = dim(e u ). T x W s (x) = E s x and T x W u (x) = E u x Closed + Bounded + Hyperbolic = Interesting Dynamics

The Structure of Hyperbolic Sets p. 16/35 Morse-Smale Diffeormophisms A diffeo. f is Morse-Smale if the only recurrent points are a finite number hyperbolic periodic points and the stable and unstable manifolds of each periodic point is transverse. so dynamics are gradient like.

The Structure of Hyperbolic Sets p. 17/35 Weak Palis Conjecture Note: Recent Theorem says horseshoes are very common for diffeomorphisms. Theorem 1 (Weak Palis Conjecture) For any smooth manifold the set of there is an open and dense set of C 1 diffeomorphisms that are either Morse-Smale or contain a horseshoe. Proof announced by Crovisier, based on work of Bonatti, Gan, and Wen.

The Structure of Hyperbolic Sets p. 18/35 Hyperbolic Toral Automorphisms Take the Matrix A = [ ] 2 1. 1 1 This matrix has det(a) = 1, one eigenvalue λ s (0, 1), and one eigenvalue λ u (1, ). So one contracting direction and one expanding direction.

The Structure of Hyperbolic Sets p. 19/35 Anosov Diffeomorphisms Since A has determinant 1 it preserves Z 2 there is induced map on torus f A from A. At every point x T 2 there is a contacting and expanding direction.

The Structure of Hyperbolic Sets p. 20/35 Anosov Diffeos in 2-dimensions In two dimensions only the torus supports Anosov diffeomorphisms and all are topologically conjugate to hyperbolic toral automorphisms. Two maps f : X X and g : Y Y are conjugate if there is a continuous homeomorphism h : X Y such that hf = gh.

The Structure of Hyperbolic Sets p. 21/35 Attractors Definition 2 A set X has an attracting neighborhood if neighborhood U of X such that X = n N fn (U).

The Structure of Hyperbolic Sets p. 21/35 Attractors Definition 3 A set X has an attracting neighborhood if neighborhood U of X such that X = n N fn (U). A hyperbolic set Λ is a hyperbolic attractor if Λ is transitive (contains a point with a dense orbit) and has an attracting neighborhood.

The Structure of Hyperbolic Sets p. 21/35 Attractors Definition 4 A set X has an attracting neighborhood if neighborhood U of X such that X = n N fn (U). A hyperbolic set Λ is a hyperbolic attractor if Λ is transitive (contains a point with a dense orbit) and has an attracting neighborhood. For a compact surface result of Plykin says there must be at least 3 holes for a hyperbolic attractor.

The Structure of Hyperbolic Sets p. 22/35 Plykin Attractor V

The Structure of Hyperbolic Sets p. 22/35 Plykin Attractor V f(v)

The Structure of Hyperbolic Sets p. 23/35 Dynamics of Attractors V is an attracting neighborhood and Λ = n N f n (V ). In two dimensions a hyperbolic attractor looks locally like a Cantor set interval. The interval is the unstable direction the Cantor set is the stable direction. Hyperbolic attractors have dense periodic points and a point with a dense orbit.

The Structure of Hyperbolic Sets p. 24/35 Locally Maximal A hyperbolic set Λ is locally maximal (or isolated) if open set U such that Λ = n Zf n (U) All the examples we looked at are locally maximal

The Structure of Hyperbolic Sets p. 25/35 Properties of Locally Maximal Sets Locally maximal transitive hyperbolic sets have nice properties including: 1. Shadowing 2. Structural Stability 3. Markov Partitions 4. SRB measures (for attractors)

The Structure of Hyperbolic Sets p. 26/35 Shadowing A sequence x 1,x 2,...,x n is an ǫ pseudo-orbit if d(f(x i ),x i+1 ) < ǫ for all 1 i < n. A point y δ-shadows an ǫ pseudo-orbit if d(f i (y),x i ) < δ for all 1 i n.

The Structure of Hyperbolic Sets p. 27/35 Shadowing Diagram f(x ) 1 x 1

The Structure of Hyperbolic Sets p. 27/35 Shadowing Diagram f(x ) x 2 1 x 1

The Structure of Hyperbolic Sets p. 27/35 Shadowing Diagram f(x ) x 2 1 x 3 f(x 2) x 1

The Structure of Hyperbolic Sets p. 27/35 Shadowing Diagram y f(x ) x 2 1 x 3 f(x 2) x 1

The Structure of Hyperbolic Sets p. 27/35 Shadowing Diagram y f(y) f(x ) x 2 1 x 3 f(x 2) x 1

The Structure of Hyperbolic Sets p. 27/35 Shadowing Diagram y f(y) f(x ) x 2 1 2 f (y) x 3 f(x 2) x 1

The Structure of Hyperbolic Sets p. 28/35 Shadowing Theorem Theorem 5 (Shadowing Theorem) Let Λ be a compact hyperbolic invariant set. Given δ > 0 ǫ,η > 0 such that if {x j } j 2 j=j 1 is an ǫ pseudo-orbit for f with d(x j, Λ) < η for j 1 j j 2, then y which δ-shadows {x j }. If j 1 = and j 2 =, then y is unique. If Λ is locally maximal and j 1 = and j 2 =, then y Λ.

The Structure of Hyperbolic Sets p. 29/35 Structural Stability Theorem 6 (Structural Stability) If Λ is a hyperbolic set for f, then there exists a C 1 open set U containing f such that for all g U there exists a hyperbolic set Λ g and homeomorphism h : Λ Λ g such that hf = gh.

The Structure of Hyperbolic Sets p. 30/35 Markov Partition Decomposition of Λ into finite number of dynamical rectangles R 1,...,R n such that for each 1 i n

The Structure of Hyperbolic Sets p. 30/35 Markov Partition Decomposition of Λ into finite number of dynamical rectangles R 1,...,R n such that for each 1 i n int(r i ) int(r j ) = if i j

The Structure of Hyperbolic Sets p. 30/35 Markov Partition Decomposition of Λ into finite number of dynamical rectangles R 1,...,R n such that for each 1 i n int(r i ) int(r j ) = if i j for some ǫ sufficiently small R i is (W u ǫ (x) R i) (W s ǫ (x) R i) x R i, f(x) R j, and i j is an allowed transition in Σ, then f(w s (x,r i )) R j and f 1 (W u (f(x),r j )) R i.

The Structure of Hyperbolic Sets p. 31/35 SRB Measurs If Λ is a hyperbolic attractor measure µ on Λ such that for a.e x in basin of attraction and any observable φ: lim n 1 n Σn i=1φ(f i (x)) = Λ φ dµ

The Structure of Hyperbolic Sets p. 32/35 Question 1 Katok and Hasselblatt: Question 1: Let Λ be a hyperbolic set... and V an open neighborhood of Λ. Does there exist a locally maximal hyperbolic set Λ such that Λ Λ V? Crovisier(2001) answers no for specific example on four torus.

The Structure of Hyperbolic Sets p. 33/35 Robust and Markov Theorems Theorem 7 (F.) On any compact manifold M, where dim(m) 2, there exists a C 1 open set of diffeomorphisms, U, such that any f U has a hyperbolic set that is not contained in a locally maximal hyperbolic set. Theorem 8 (F.) If Λ is a hyperbolic set and V is a neighborhood of Λ, then there exists a hyperbolic set Λ with a Markov partition such that Λ Λ V.

The Structure of Hyperbolic Sets p. 34/35 Hyperbolic Sets with Interior Theorem 9 (F.) If Λ is a hyperbolic set with nonempty interior, then f is Anosov if 1. Λ is transitive 2. Λ is locally maximal and M is a surface

The Structure of Hyperbolic Sets p. 35/35 Hyperbolic Attractors on Surfaces Theorem 10 (F.) If M is a compact smooth surface, Λ is a hyperbolic attractor for f, and hyperbolic for g, then Λ is either a hyperbolic attractor or repeller for g. So if we know the topology of the set and we know that it is hyperbolic we know it is an attractor. A set Λ is a repeller if there exists neighborhood V such that Λ = n N f n (V ).