Entropy of C 1 -diffeomorphisms without dominated splitting

Transcription

1 Entropy of C 1 -diffeomorphisms without dominated splitting Jérôme Buzzi (CNRS & Université Paris-Sud) joint with S. CROVISIER and T. FISHER June 15, 2017 Beyond Uniform Hyperbolicity - Provo, UT

2 Outline Introduction Understanding topological entropy Topological and measured entropies Creating a horseshoe Localized perturbative theorem Ingredients of the proof Dissipative diffeomorphisms Horseshoe entropy Infinitely many homoclinic classes Conservative diffeomorphisms Entropy formulas Instability, continuity,... Borel classification Conclusion

3 Understanding topological entropy f : M M compact, connected, without boundary Topological Entropy (Adler-McAndrew-Konheim 1968) h top : Diff r (M) [0, ) How does it vary? - continuity: Misiurewicz,Katok (low d); Newhouse,Yomdin C - local constancy: stability beyond hyperbolicity (B-Fisher) - robust instability (diffeos not approximated by local constancy)? What are its sources? - homology: Shub s Entropy conjecture - volume growth: Yomdin, Newhouse C - combinatorics through Markov partitions: Bowen - combinatorics through horseshoes: Katok Diff 1+α (M 2 ) How does it classify? - Generators: Jewett-Krieger, Hochman, Burguet-Downarowicz - Almost... conjugacy: Adler-Marcus, Boyle-B-Gomez Which values does it takes?

4 Topological and measure entropies - Definitions f : M M C 0, compact, µ P erg (f ) Topological Entropy (Adler-McAndrew-Konheim 1965; Bowen 1971) h top (f ) = lim ɛ 0+ h top (f, ɛ) h top (f, ɛ) = lim sup n 1 n log r f (ɛ, n, M) Measured Entropy (Kolmogorov-Sinai 1958; Katok 1980) h(f, µ) = lim ɛ 0+ h(f, µ, ɛ) h(f, µ, ɛ) = lim sup n 1 n log r f (ɛ, n, µ) Tail Entropy (Misiurewicz-Bowen 1973) h (f ) = lim ɛ 0+ h (f, ɛ) h 1 (f, ɛ) = sup h top(f, B f (x, ɛ, )) = lim lim sup x M δ 0 n Variational principle (Goodman 1971) h top (f ) = sup{h(f, µ) : µ P erg (f )} Measure maximizing the entropy (mme) n sup x M r f (δ, n, B f (x, ɛ, n)) µ max P erg (f ) with h(f, µ max ) = sup{h(f, µ) : µ P erg (f )} Newhouse: C = existence

5 Main Theorem For µ P erg (f ), f Diff 1 (M), M closed Lyapunov Exponents λ 1 (f, µ) λ 2 (f, µ) λ d (f, µ) Ruelle s inequality: h(f, µ) (f, µ) := min ( i λ i(f, µ) +, i λ i(f, µ) ) Main Theorem (B-Crovisier-Fisher) U a neighborhood of f in Diff 1 (M), Diff 1 vol(m) or Diff 1 ω(m) O periodic orbit with large period and no strong dominated splitting Then, for each U O, there is a horseshoe O K U for g U s.t. h top (g, K) (g, O) = (f, O) Moreover: {g f } U \ O; can preserve a homoclinic relation Remark Optimal : lim sup g f h top (g) sup µ (f, µ) Remark Specific to C 1 -topology: tools; factor 1/r in C r Locally uniform bounds on required period, domination - Newhouse 1978 (d = 2); - Catalan-Tahzibi 2014 (symplectic, entropy min i ( λ i (f, O) )) (see also: Catalan 2016)

6 Tools for localized perturbations - Perturbations of periodic linear cocycles (Bochi-Bonatti, Gourmelon, new for symplectic) - making spectrum simple with rational angles; - mixing the stable (unstable) exponents - creating a small angle - Local support with homoclinic connection, form or symplectic form (Gourmelon): - Franks Lemma with linearization (Avila for volume-preserving) - Homoclinic tangency from lack of domination (Gourmelon) Localized perturbations of conservative C 1 diffeomorphisms, arxiv:1612:06914

7 Proof - Part I: circular permutation f Diff 1 (M) with O(p) a long periodic orbit with weak domination Use previous tools to create, by perturbations: 1. Transverse homoclinic point and locally linear horseshoe K 0 p 2. Point x K 0 with large period and Df π(x) x = Λ s Id E s Λ u Id E u 3. Homoclinic tangency z for O(x) 4. Linearize around O(x) so loc. invariant E 1... E }{{ k } E s 5. With F = T z W s (x) T z W u (x) assumed to be 1-d: T z W s (x) = F F s 1 F s k 1 T z W u (x) = F u 1 F u k d 1 F T z M = F (F s 1 F s k 1 ) G (F u 1 F u k d 1 ) E k+1... E d }{{} E u 6. Using Df π(x) =homothety homothety and F F u 1 F u k d 1 E s : Perturb future of z to get F E 1, Fi s E i+1 and G E k+1, Fj u E k+1+j Similarly in the past Fi s E i, G E k, Fj u E k+j, F E d Conclusion τ Wloc u (x) f m Wloc s (x) s.t. Df m τ.e i = E i+1 (E d+1 = E 1 )

8 Proof - Part II: entropy from exponents Case d = 3, k = 2 (λ 1 < λ 2 < 0 < λ 3 ) g n+m e3 g l+n e1 e2 δ 1 = δ, δ 2 = e nλ1, δ 3 = δe λ3 n Image by f n has height along e 1 : δe (λ1+λ2)n Wiggles (x 1,..., x d ) (x 1,..., x k, x k+1 + H cos(πnx 1 ), x k+2,..., x d ) - to cross: H C(e (λ1+λ2)n + e λ3n )δ - to be C 1 -small: N = o(h 1 ) Entropy log N n max( λ 1 λ 2, λ 3 ) = (f, O(x))

9 Application 1: C 1 horseshoes from LACK of domination Let f Diff 1 (M) be generic (ie, belonging to dense G δ in Diff 1 (M)) Theorem (B-Crovisier-Fisher) For any µ P erg (f ), if suppµ has no dominated splitting, then are horseshoes K n approximating µ: (i) in entropy; (ii) in Hausdorff distance; (iii) in weak-star topology Compare Katok C 1+ ; Gan, Gelfert C 1 + adapted dominated splitting Remark Does not say that K n supp(µ) or homoclinically related Ingredients of the proof: - Ergodic closing lemma with control of exponents - Main theorem

10 Application 2: Infinitely many homoclinic classes Homoclinic relation and classes for hyperbolic periodic orbit O: O O W s (O) W u (O ) et W u (O) W s (O ) HC(O) := O O O compact, invariant, transitive Theorem (B-Crovisier-Fisher) f Diff 1 (M) generic Any HC(O) without dominated splitting is accumulated by infinitely many homoclinic classes with entropy bounded away from zero More precisely, lim inf n h top (HC(O n )) sup O O (f, O ) Remark Newhouse s theorem would suffice (smaller bound) Ingredients: - O O with (O ) > (O), long period, weak domination - Franks Lemma and linear perturbation to make O a sink/source - undo the perturbation inside the basin - Main Theorem

11 Application 3: Entropy formulas in conservative settings M closed manifold with dim d 2, ω volume or symplectic form Let E 1 ω(m) := int({f Diff 1 ω(m) : no domination}) Theorem (B-Crovisier-Fisher) The topological entropy of a generic f E 1 ω(m) is equal to: (1) sup{h top (f, K) : K horseshoe} (2) sup{ (f, O) : O periodic orbit} (3) max 0<k<d lim n 1 n log sup E G k (TM) Jac(f n, E) generalizes, strengthens Catalan-Tahzibi (2014) Ingredients of the proof - sup K h top (K) h top (f ) = sup µ h(µ) sup µ (µ) (always) - erg. measures arb. dense periodic orbits (generic, Abdenur-Bonatti-Crovisier) sup µ (µ) = (f ) := sup O (f, O) - is continuous at generic diffeo - sup K h top (K) > (f ) ɛ open and dense

12 Application 4: Instability of the entropy M closed manifold with dim d 2, ω volume or symplectic form Let E 1 ω(m) := int({f Diff 1 ω(m) : no domination}) Theorem (B-Crovisier-Fisher) The topological entropy of a generic f E 1 ω(m) is equal to: (1) sup{h top (f, K) : K horseshoe} (2) sup{ (f, O) : O periodic orbit} (3) max 0<k<d lim n 1 n log sup E G k (TM) Jac(f n, E) Corollary h top is nowhere locally constant in E 1 ω(m) (robust instability) Corollary For any dense G δ G E 1 ω(m), h top (G) uncountable Corollary Generic f E 1 ω(m) is a continuity point of h top Diff 1 ω(m) Corollary C 1 generically : no domination h (f ) = h top (f )

13 Application 5: No mme and Borel classification M closed manifold with dim d 2, ω volume or symplectic form Theorem (B-Crovisier-Fisher) Generic f E 1 ω(m) has no measure maximizing the entropy Remark The diffeos with m.m.e. are dense (Newhouse theorem for C ) Combining horseshoes, no m.m.e. and Hochman (arxiv 2015): Corollary (B-C-F) There is dense G δ subset of E 1 ω(m) among which the topological entropy is a complete invariant for Borel conjugacy after removing periodic points

14 Proof of no m.m.e. - Concentration phenomenon f E 1 ω(m), dim M 2 Dynamical ball for x M, ɛ > 0: B f (x, ɛ, n) := {y M : 0 k < n d(f k y, f k x) < ɛ} Proposition 0 < ɛ, α < 1, for a dense set of f 0 E 1 ω(m), δ > 0, finite X M s.t. (*) if f close to f 0, µ P erg (f ), and h(f, µ) > h top (f ) δ, then µ( x X B f (x, ɛ, #X )) > 1 α Proof of Theorem. 1) G dense G δ E 1 ω(m) s.t. 0 < ɛ, α < 1 f G δ > 0 X finite satisfying (*) 2) Let f G, µ m.m.e., and ɛ > 0. From Katok s formula, need to bound: r f (µ, ɛ, n) := min{#c : µ( x C B f (x, ɛ, n)) > 1/2} 3) Take 0 < α << 1/ log min{#c : x C B(x, ɛ) = M}. Apply (*)

15 Conclusion Conjecture (higher smoothness) Given a C r -diffeo with hyperbolic periodic point p in a cycle of basic sets (see Gourmelon) with no dominated splitting, there is a C r -perturbation with a horseshoe with entropy (p)/r Question (internal perturbations) For a homoclinic class of a C 1 -generic diffeo, is the topological entropy the supremum of that of the horseshoes it contains? Question (entropy instability) Show that {f Diff 1 (M) : h top not locally constant at f } has non-empty interior Problem (entropy stability) Characterize the locus of entropy stability {U open in Diff 1 ω (M) : h top U = const} (Generically in Diff 1 ω (M): no domination h = h top) arxiv: , arxiv:1612:06914 JB, S. Crovisier, T. Fisher, The entropy of C 1 -diffeomorphisms without a dominated splitting JB, S. Crovisier, T. Fisher, Local perturbations of conservative C 1 diffeomorphisms