The Structure of Hyperbolic Sets

Transcription

1 The Structure of Hyperbolic Sets p. 1/35 The Structure of Hyperbolic Sets Todd Fisher Department of Mathematics University of Maryland, College Park

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

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

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

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

6 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.

7 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.

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

9 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.

10 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. For instance, say f(x,y) = [ 1/ ] [ ] x y

11 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 }.

12 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 }.

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

14 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).

15 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). P

16 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.

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

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

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

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

21 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

22 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:

23 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: n f (D) D p x

24 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 R 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

25 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)

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

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

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

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

30 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)

31 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) So Horseshoe is very interesting dynamically.

32 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

33 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

34 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.

35 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.

36 The Structure of Hyperbolic Sets p. 18/35 Hyperbolic Toral Automorphisms Take the Matrix A = [ ] 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.

37 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.

38 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. A diffeomorphism is Anosov if the entire manifold is a hyperbolic set. So f A is Anosov.

39 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.

40 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).

41 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.

42 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.

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

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

45 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.

46 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

47 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)

48 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.

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

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

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

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

53 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

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

55 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.

56 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.

57 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

58 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

59 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)

60 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.

61 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

62 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?

63 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.

64 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. Related questions: 1. Can this be robust? 2. Can this happen on other manifolds, in lower dimension, on all manifolds?

65 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.

66 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

67 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 ).