Symbolic dynamics and non-uniform hyperbolicity

Transcription

1 Symbolic dynamics and non-uniform hyperbolicity Yuri Lima UFC and Orsay June, 2017 Yuri Lima (UFC and Orsay) June, / 83

2 Lecture 1 Yuri Lima (UFC and Orsay) June, / 83

3 Part 1: Introduction Yuri Lima (UFC and Orsay) June, / 83

4 Cat map f = [ ] T2 T 2 Yuri Lima (UFC and Orsay) June, / 83

5 Cat map wrt x, y axis Yuri Lima (UFC and Orsay) June, / 83

6 Cat map wrt eigendirections (Adler-Weiss, Sinai) Yuri Lima (UFC and Orsay) June, / 83

7 The Markov property Rectangles satisfy a crossing property: R S ALLOWED: f f (R) R S NOT ALLOWED: f f (R) Yuri Lima (UFC and Orsay) June, / 83

8 Symbolic model for f V = {rectangles} E = {R S f (R) S /= } Σ = {Z indexed paths on G} σ Σ Σ left shift π Σ T 2, {π(..., R 0,...)} = f n R n f n R n n 0 Yuri Lima (UFC and Orsay) June, / 83

9 Symbolic model for f π Σ T 2, {π(..., R 0,...)} = f n R n f n R n n 0 Markov property intersection Expansion/contraction intersection is a singleton Yuri Lima (UFC and Orsay) June, / 83

10 Conclusion f and σ are the same : Σ σ π Σ π T 2 f T 2 Yuri Lima (UFC and Orsay) June, / 83

11 Conclusion Simple iteration Counting of periodic points Ergodic properties of inv. measures, e.g. measures of maximal entropy Yuri Lima (UFC and Orsay) June, / 83

12 Symbolic model for diffeomorphisms G = (V, E) oriented graph with V countable Σ = {Z indexed paths on G} σ Σ Σ left shift (Σ, σ) = topological Markov shift (TMS) π Σ M coding (Σ, σ, π) symbolic model Yuri Lima (UFC and Orsay) June, / 83

13 Symbolic model for flows (Σ, σ) a TMS r Σ R positive Σ r = {(v, t) v Σ, 0 t r(v)} with identification (v, r(v)) (σ(v), 0) σ r Σ r Σ r unit speed vertical flow graph(r) (v, r(v)) Σ (v, 0) (σ(v), 0) (Σ r, σ r ) = topological Markov flow (TMF) π r Σ r M coding (Σ r, σ r, π r ) symbolic model Yuri Lima (UFC and Orsay) June, / 83

14 Symbolic models for uniformly hyperbolic systems Adler-Weiss 1967: 2 dim hyperbolic toral automorphisms Sinai 1968: Anosov diffeomorphisms Bowen 1970: Axiom A diffeomorphisms Ratner 1973: Anosov flows Bowen 1973: Axiom A flows These include: Hyperbolic toral automorphisms Geodesic flows on manifolds with negative sectional curvature Yuri Lima (UFC and Orsay) June, / 83

15 Part 2: Non-uniform hyperbolicity Yuri Lima (UFC and Orsay) June, / 83

16 Non-uniformly hyperbolic systems (NUH) Asymptotic hyperbolicity f M M or ϕ = {ϕ t } t R M M χ hyperbolic measure: invariant measure s.t. Lyapunov exp. > χ, for all non-trivial Lyapunov exp. These include: f M 2 M 2 diffeomorphism with positive entropy: µ ergodic and h µ > χ µ is χ hyperbolic (Ruelle s inequality) ϕ M 3 M 3 flow with positive entropy, e.g. geodesic flows on surfaces with nonpositive curvature Yuri Lima (UFC and Orsay) June, / 83

17 Some positive curvature, yet χ hyperbolic Donnay 88, Burago 92, Burns-Donnay 97 Yuri Lima (UFC and Orsay) June, / 83

18 Some positive curvature, yet χ hyperbolic Donnay 88, Burago 92, Burns-Donnay 97 Yuri Lima (UFC and Orsay) June, / 83

19 Some positive curvature, yet χ hyperbolic Donnay 88 Yuri Lima (UFC and Orsay) June, / 83

20 Some positive curvature, yet χ hyperbolic Donnay 88 Yuri Lima (UFC and Orsay) June, / 83

21 NUH expanding maps Interval maps with positive entropy: Viana maps: (θ, x) T [0, 1] (dθ, a + ε sin(2πθ) x 2 ) Yuri Lima (UFC and Orsay) June, / 83

22 NUH billiards Sinai s billiard table Bunimovich billiard tables Yuri Lima (UFC and Orsay) June, / 83

23 Part 3: Results Yuri Lima (UFC and Orsay) June, / 83

24 Symbolic dynamics for NUH surface diffeomorphisms... Katok 1980: surface diffeo with h > 0 has horseshoes of large topological entropy. Theorem (Sarig 2013) Let f be a C 1+β surface diffeomorphism. For all χ > 0 there is (Σ, σ) TMS and π Σ M Hölder continuous s.t.: (1) π σ = f π. (2) π[σ # ] has full measure for all χ hyperbolic measures. (3) For all x π[σ # ], it holds {v Σ # π(v) = x} <. Σ # = {v Σ v, w V s.t. v n = v for infinitely many n > 0 v n = w for infinitely many n < 0 }. Yuri Lima (UFC and Orsay) June, / 83

25 ... and for any dimension Theorem (Ben-Ovadia 2016) Let f M M be a C 1+β diffeomorphism, where M is a closed smooth Riemannian manifold of any dimension. Then the conclusion of Sarig s theorem holds. Yuri Lima (UFC and Orsay) June, / 83

26 Symbolic dynamics for NUH 3 dim flows M = 3 dim closed smooth Riemannian manifold X = C 1+β vector field on M s.t. X 0 everywhere ϕ = flow generated by X µ = χ hyperbolic measure (fixed!) Theorem (L-Sarig 2014) There is (Σ r, σ r ) TMF and π r Σ r M Hölder continuous s.t.: (1) π r σr t = ϕ t π r for all t R. (2) π r [Σ # r ] has full measure µ measure. (3) For all x π r [Σ # r ], it holds {(v, t) Σ # r π r (v, t) = x} <. Σ # r = {(v, t) Σ r v Σ # }. Yuri Lima (UFC and Orsay) June, / 83

27 Symbolic dynamics for NUH billiards T R 2 compact domain with piecewise smooth boundary f = billiard map on T dµ = cos θdrdθ, a f invariant Liouville measure Theorem (L-Matheus) If µ is ergodic and h > 0 then there is (Σ, σ) TMS and π Σ M Hölder continuous s.t.: (1) π σ = f π. (2) π[σ # ] has full µ measure. (3) For all x π[σ # ], it holds {v Σ # π(v) = x} <. Yuri Lima (UFC and Orsay) June, / 83

28 NUH billiards Sinai s billiard table Bunimovich billiard tables Yuri Lima (UFC and Orsay) June, / 83

29 Non-uniformy expanding interval maps f [0, 1] [0, 1] piecewise C 1+β S = {discontinuities} {flat points} Near S : f, f polynomial-like Inverse branches of f of polynomial-like radii µ invariant measure is f adapted if: µ{orbits converging to S exponentially fast} = 0 Yuri Lima (UFC and Orsay) June, / 83

30 Non-uniformy expanding interval maps f [0, 1] [0, 1] natural extension [0, 1] f [0, 1] [0, 1] f [0, 1] µ = lift of µ to [0, 1] Theorem (L) Let f be as above. For all χ > 0 there is (Σ, σ) TMS and π Σ [0, 1] Hölder continuous s.t.: (1) π σ = f π. (2) π[σ # ] has full µ measure for all χ hyperbolic f adapted measures µ. (3) For all x π[σ # ], it holds {v Σ # π(v) = x} <. Yuri Lima (UFC and Orsay) June, / 83

31 Applications Let f be a C 1+β surface diffeomorphism with h = h top (f ) > 0 Sarig 2013: p s.t. Per np (f ) const e nph Sarig 2013: at most countably many ergodic mme s Buzzi-Crovisier-Sarig 2017: if f is C and transitive, then! mme Sarig 2011: if µ is ergodic equilibrium state of Hölder potential s.t. h µ (f ) > 0 then (f, µ) is either Bernoulli or Bernoulli finite rotation Boyle-Buzzi 2014: almost Borel structure of surface diffeomorphisms Backes-Poletti-Varandas 2016: simplicity of Lyapunov spectrum over NUH systems Let ϕ as in L-Sarig 2015 with h = h top (ϕ) > 0 L-Sarig 2015: Per T (ϕ) const eth T L-Sarig 2015: at most countably many ergodic mme s Ledrappier-L-Sarig 2016: if µ is ergodic equilibrium state of Hölder potential s.t. h µ (ϕ) > 0 then (ϕ, µ) is either Bernoulli or Bernoulli rotational flow Yuri Lima (UFC and Orsay) June, / 83

32 Part 4: Other directions Yuri Lima (UFC and Orsay) June, / 83

33 Other directions NUH endomorphisms (L) NUH flow: 1. with zeroes 2. incomplete 3. code all χ hyperbolic measures simultaneously Buzzi-Crovisier-L: 3 dim flows with no zeroes code all χ hyperbolic measures simultaneously Yuri Lima (UFC and Orsay) June, / 83

34 Lecture 2 Yuri Lima (UFC and Orsay) June, / 83

35 The theorem of Sarig Σ # = {v Σ v, w V s.t. v n = v for infinitely many n > 0 v n = w for infinitely many n < 0 } Theorem (Sarig 2013) Let f be a C 1+β surface diffeomorphism. For all χ > 0 there is (Σ, σ) TMS and π Σ M Hölder continuous s.t.: (1) π σ = f π. (2) π[σ # ] has full measure for all χ hyperbolic measures. (3) There exists a function N {vertices of Σ} N satisfying the following: if x = π(v) with v n = v for infinitely many n > 0 and v n = w for infinitely many n < 0, then {v Σ # π(v) = x} < N(v)N(w). Yuri Lima (UFC and Orsay) June, / 83

36 Part 5: The method of pseudo-orbits Yuri Lima (UFC and Orsay) June, / 83

37 The method of pseudo-orbits (Bowen) Four steps (1) Lyapunov charts Ψ x : if f (x) y, then f xy = Ψ 1 y f Ψ x [ A 0 0 B ] (2) Coarse graining: X finite and δ dense in M. (3) Infinite-to-one extension: V = {Ψ x x X } and E = {Ψ x Ψ y f (x) y, f 1 (y) x} Σ = Σ(V, E), v Σ is a pseudo-orbit π Σ M defined by graph transforms: π(v) = V s [v] V u [v] (4) Bowen-Sinai refinement: Z = {Z v v V }, where Z v = {π(v) v Σ, v 0 = v} Z, Z Z E Z,Z = partition of Z into four pieces R = refinement of all E Z,Z R defines a finite-to-one symbolic model Yuri Lima (UFC and Orsay) June, / 83

38 Method of pseudo-orbits for NUH surface diffeos (Sarig) Four steps (1) Pesin charts Ψ x : if ε overlap, then f xy = Ψ 1 y f Ψ x [ A 0 0 B ] (2) Coarse graining: X countable and dense in {Pesin charts} (3) Infinite-to-one extension: V = X and E = {Ψ x Ψ y ε overlap} Σ = Σ(V, E), v Σ is a pseudo-orbit π Σ M defined by graph transforms (strong assumption on s curves): π(v) = V s [v] V u [v] (4) Bowen-Sinai refinement: Z = {Z v v V }, where Z v = {π(v) v Σ, v 0 = v} Z, Z Z E Z,Z = partition of Z into four pieces R = refinement of all E Z,Z Yuri Lima (UFC and Orsay) June, / 83

39 Problem R can be uncountable Yuri Lima (UFC and Orsay) June, / 83

40 (4) Bowen-Sinai refinement: Z = {Z v v V }, where Z v = {π(v) v Σ #, v 0 = v} Z, Z Z E Z,Z = partition of Z into four pieces R = refinement of all E Z,Z Z locally finite R is countable Yuri Lima (UFC and Orsay) June, / 83 Method of pseudo-orbits for NUH surface diffeos (Sarig) Five steps (1) ε double charts Ψ ps,p u x = (Ψ x [ p s,p s ] 2, Ψ x [ p u,p u ] 2) (2) Coarse graining: X countable and dense in {ε double charts} (3) Infinite-to-one extension: V = X and E = {Ψ ps,p u x Ψ qs,q u y ε overlap and maximality of p s, p u, q s, q u } Σ = Σ(V, E), v Σ is an ε generalized-pseudo-orbit (ε gpo) π Σ M defined by graph transforms (strong assumption on s curves): π(v) = V s [v] V u [v] (3 1 2 ) Inverse problem: If π(v) = π(w) with v, w Σ# then v w: v = {Ψ ps n,pn u x n } n Z and w = {Ψ qs n,qn u y n } n Z x n y n, pn s qn, s pn u qn. u

41 Part 6: Hyperbolicity parameters Yuri Lima (UFC and Orsay) June, / 83

42 Hyperbolicity parameters I Take x M s.t. e s, e u distinct s.t. e s contracts in the future and expands in the past e u expands in the future and contracts in the past Angle parameter: α(x) = (e s, e u ) s parameter: s(x) = 1/2 2 ( e 2χn df n e s 2 ) n 0 u parameter: u(x) = 1/2 2 ( e 2χn df n e u 2 ) n 0 Yuri Lima (UFC and Orsay) June, / 83

43 The set NUH NUH = {x M s(x), u(x) < } Yuri Lima (UFC and Orsay) June, / 83

44 Hyperbolicity parameters II Size of Pesin chart: Q(x) = sin α(x) s(x) 2 + u(x) 2 Large power Size of stable manifold: p s (x) = inf{e εn Q(f n (x)) n 0}. Size of unstable manifold: p u (x) = inf{e εn Q(f n (x)) n 0}. Yuri Lima (UFC and Orsay) June, / 83

45 Explanation of ε double charts Ψ ps,p u x p s p s (x), p u p u (x) Yuri Lima (UFC and Orsay) June, / 83

46 Summary e s, e u α(x) s(x) u(x) Q(x) p s (x) p u (x) NUH Yuri Lima (UFC and Orsay) June, / 83

47 Summary e s, e u α(x) s(x) u(x) Q(x) p s (x) p u (x) NUH Directions of contraction/expansion Yuri Lima (UFC and Orsay) June, / 83

48 Summary e s, e u α(x) s(x) u(x) Q(x) p s (x) p u (x) NUH Directions of contraction/expansion Distinction between directions Yuri Lima (UFC and Orsay) June, / 83

49 Summary e s, e u α(x) s(x) u(x) Q(x) p s (x) p u (x) NUH Directions of contraction/expansion Distinction between directions Weak measurement of contraction Yuri Lima (UFC and Orsay) June, / 83

50 Summary e s, e u α(x) s(x) u(x) Q(x) p s (x) p u (x) NUH Directions of contraction/expansion Distinction between directions Weak measurement of contraction Weak measurement of expansion Yuri Lima (UFC and Orsay) June, / 83

51 Summary e s, e u α(x) s(x) u(x) Q(x) p s (x) p u (x) NUH Directions of contraction/expansion Distinction between directions Weak measurement of contraction Weak measurement of expansion Size of Pesin chart Yuri Lima (UFC and Orsay) June, / 83

52 Summary e s, e u α(x) s(x) u(x) Q(x) p s (x) p u (x) NUH Directions of contraction/expansion Distinction between directions Weak measurement of contraction Weak measurement of expansion Size of Pesin chart Size of stable manifold Yuri Lima (UFC and Orsay) June, / 83

53 Summary e s, e u α(x) s(x) u(x) Q(x) p s (x) p u (x) NUH Directions of contraction/expansion Distinction between directions Weak measurement of contraction Weak measurement of expansion Size of Pesin chart Size of stable manifold Size of unstable manifold Yuri Lima (UFC and Orsay) June, / 83

54 Summary e s, e u α(x) s(x) u(x) Q(x) p s (x) p u (x) NUH Directions of contraction/expansion Distinction between directions Weak measurement of contraction Weak measurement of expansion Size of Pesin chart Size of stable manifold Size of unstable manifold s, u < Yuri Lima (UFC and Orsay) June, / 83

55 Lecture 3 Yuri Lima (UFC and Orsay) June, / 83

56 Part 7: The inverse problem Yuri Lima (UFC and Orsay) June, / 83

57 The inverse problem Fix x M: If π(v) = x, what is v? Yuri Lima (UFC and Orsay) June, / 83

58 π(v) = x R 2 M x 1 f 1 (x) x 0 x x 1 f (x) Yuri Lima (UFC and Orsay) June, / 83

59 π(v) = x R 2 M x 1 f 1 (x) x 0 x x 1 f (x) Yuri Lima (UFC and Orsay) June, / 83

60 e s, e u are well-defined at x R 2 M x 1 f 1 (x) x 0 x x 1 f (x) Yuri Lima (UFC and Orsay) June, / 83

61 Contraction/expansion along e s /e u R 2 M x 1 f 1 (x) x 0 x x 1 f (x) At x 0 : contraction/expansion at least e χ At x: contraction/expansion at least e χ/2 Yuri Lima (UFC and Orsay) June, / 83

62 α(x) α(x 0 ) π 2 α(x) Yuri Lima (UFC and Orsay) June, / 83

63 Control of s, u Goals: s(x), u(x) < s(x) s(x 0 ), u(x) u(x 0 ) Yuri Lima (UFC and Orsay) June, / 83

64 The graph transform improves ratios x 0 x x 1 f (x) s(x) s(x 0 ) is better than s(f (x)) s(x 1 ) Yuri Lima (UFC and Orsay) June, / 83

65 The graph transform improves ratios x 0 x x 1 f (x) s(x) 2 = 1 + Cs(f (x)) 2 s(x 0 ) Cs(x 1 ) 2 If s(f (x))2 s(x 1 ) 2 = K 1 then s(x) 2 s(x 0 ) Cs(x)2 1 + Cs(x 1 ) 2 = 1 + KCs(x 1) 2 < [Term < 1] K 1 + Cs(x 1 ) 2 [Term < 1] = exp[ Q(x 0 ) β/4 ] v Σ # infinitely many improvements Yuri Lima (UFC and Orsay) June, / 83

66 If v Σ #, then control of s, u, Q, p s, p u The graph transform improves ratios + s(x), u(x) < s(x) s(x 0 ), u(x) u(x 0 ) Q(x) Q(x 0 ), p s (x) p s 0, pu (x) p u 0 Yuri Lima (UFC and Orsay) June, / 83

67 Projections of measures Σ π M ν = σ invariant measure on Σ Poincaré recurrence ν supported on Σ # µ = ν π 1 supported on NUH. Yuri Lima (UFC and Orsay) June, / 83

68 Part 8: Finite-to-one extension Yuri Lima (UFC and Orsay) June, / 83

69 Review Σ = Σ(V, E), V = countably many ε double charts π Σ M defined by graph transforms Inverse theorem: x M, the equation π(v) = x, v Σ # has a unique solution up to bounded error. Z = {Z v v V }, where Z v = {π(v) v Σ #, v 0 = v}: Inverse theorem Z is locally finite Symbolic structure of Σ Z has symbolic Markov property Yuri Lima (UFC and Orsay) June, / 83

70 Markov partition R R = refinement of Z Local finiteness: R R, #{Z Z Z R} < Z Z, #{R R R Z} < R has geometrical Markov prop. Yuri Lima (UFC and Orsay) June, / 83

71 New TMS Σ = Σ( V, Ê) V = R R S if f (R) S /= Yuri Lima (UFC and Orsay) June, / 83

72 New coding π Σ M π(r) = f n (R n ) f n (R n ) n 0 Yuri Lima (UFC and Orsay) June, / 83

73 New coding π Σ M π(r) = f n (R n ) f n (R n ) n 0 Yuri Lima (UFC and Orsay) June, / 83

74 Relation between π and π π(r) = x Z v n Z v0 Z vn R 0 R n R n x f n (x) f n (x) v Σ s.t. { Z v n R n, n Z, π(v) = x Yuri Lima (UFC and Orsay) June, / 83

75 Relation between π and π π(r) = x Z v n Z v0 Z vn R 0 R n R n x f n (x) f n (x) v Σ s.t. { Z v n R n, n Z, π(v) = x. Yuri Lima (UFC and Orsay) June, / 83

76 Injectivity of π π(r) = x = π(s) Z v n Z v0 Z vn R 0 R n R n x f n (x) f n (x) Yuri Lima (UFC and Orsay) June, / 83

77 Injectivity of π π(r) = x = π(s) Z w n Z w0 Z wn S n S n S 0 f n (x) x f n (x) Yuri Lima (UFC and Orsay) June, / 83

78 Affiliation Z R S Z Definition: R S if { Z R Z S s.t. Z Z. Yuri Lima (UFC and Orsay) June, / 83

79 The theorem of Sarig Theorem (Sarig 2013) Let f be a C 1+β surface diffeomorphism. For all χ > 0 there is ( Σ, σ) TMS and π Σ M Hölder continuous s.t.: (1) π σ = f π. (2) π[ Σ # ] has full measure for all χ hyperbolic measures. (3) There exists a function N {vertices of Σ} N satisfying the following: if x = π(r) with R n = R for infinitely many n > 0 and R n = S for infinitely many n < 0, then #{R Σ # π(r) = x} N(R)N(S). Yuri Lima (UFC and Orsay) June, / 83

80 Definition of the function N N(R) = #{(S, Z ) R Z R S and Z S}. Local finiteness N(R) < Yuri Lima (UFC and Orsay) June, / 83

81 Finiteness of π If #{R Σ # π(r) = x} > N(R)N(S) then two of them have same affiliation configuration in far past and far future, i.e. same (R n, v n ) and (R m, v m ): Z v n+1 Z vm 1 Z v n Z vm R n R m Z w n+1 Z wm 1 Diamond argument of Bowen contradiction Yuri Lima (UFC and Orsay) June, / 83

82 Summary Construction of coding: Pesin charts Generalized pseudo-orbits Inverse theorem Bowen-Sinai refinement Some properties of coding: Image: Pesin stable/unstable manifolds with hyperbolicity > χ 2 Image improved: hyperbolicity χ Projection/lift of measures: same entropy Finiteness-to-one: combinatorial description Yuri Lima (UFC and Orsay) June, / 83

83 Yuri Lima (UFC and Orsay) June, / 83