The $C^{1+\alpha }$ hypothesis in Pesin Theory revisited

Bonatti, Christian; Crovisier, Sylvain; Shinohara, Katsutoshi

doi:10.3934/jmd.2013.7.605

Cited by 20 publications

(20 citation statements)

References 14 publications

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“…for each p ∈ supp(µ 1 ), whose existence is guaranteed by Theorem 2 in [3] we cited before; in particular, h µ 1 ( f 1 ) = 0. (M 2 , f 2 ) is a two-dimensional hyperbolic torus automorphism induced by some matrix A ∈ SL(2, Z) and µ 2 is the Lebesgue measure on M 2 .…”

Section: Introductionmentioning

confidence: 76%

“…The unstable set W u (x, f ) is simply defined as the stable set of x for f −1 . In [3,Theorem 2] the authors proved the following statement.…”

Section: Introductionmentioning

confidence: 98%

“…Pugh's counter-example is about the non-existence of invariant manifolds for a single orbit of a concrete C 1 diffeomorphism. More recently, Bonatti, Crovisier and Shinohara [3] showed that nonexistence of invariant manifolds is a generic phenomenon in the C 1 category. More precisely, for a diffeomorphism f : M → M on some Riemannian manifold M, let d be the metric on M induced by its Riemannian structure.…”

Section: Introductionmentioning

confidence: 99%

“…According to [3,Remark 3], for each f ∈ R in the above Theorem, the dynamics of f on supp(µ) is a generalized adding machine (also called odometer or solenoid, see for example [5] or [22] for the definition); therefore, it is uniquely ergodic and the measure-theoretic entropy h µ ( f ) = 0. Then a natural question arises: given f ∈ Diff 1 (M) and f -invariant µ with h µ ( f ) > 0, what can we say about the structure of W s (x, f ) and W u (x, f ) for µ-a.e.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Local stable and unstable sets for positive entropy $C^1$ dynamical systems

Feng¹,

Gao²,

Huang³

et al. 2018

Preprint

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For any C 1 diffeomorphism on a smooth compact Riemannian manifold that admits an ergodic measure with positive entropy, a lower bound of the Hausdorff dimension for the local stable and unstable sets is given in terms of the measure-theoretic entropy and the maximal Lyapunov exponent. The mainline of our approach to this result is under the settings of topological dynamical systems, which is also applicable to infinite dimensional C 1 dynamical systems.

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Section: Introductionmentioning

confidence: 76%

“…The unstable set W u (x, f ) is simply defined as the stable set of x for f −1 . In [3,Theorem 2] the authors proved the following statement.…”

Section: Introductionmentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Local stable and unstable sets for positive entropy $C^1$ dynamical systems

Feng¹,

Gao²,

Huang³

et al. 2018

Preprint

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“…The classical non-uniformly hyperbolic theory always assumes that the derivative of a diffeomorphism is Hölder continuous, which could not be removed in general [4,15]. However, if the Hölder condition is replaced by the domination property between Oseledets decompositions E s and E u , some properties such as stable manifold theory and the Pesin entropy formula could still be preserved [1,19].…”

Section: Resultsmentioning

confidence: 99%

Approximation of Bernoulli measures for non-uniformly hyperbolic systems

Li¹,

Sun²,

Vargas³

et al. 2018

Ergod. Th. Dynam. Sys.

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An invariant measure is called a Bernoulli measure if the corresponding dynamics is isomorphic to a Bernoulli shift. We prove that for $C^{1+\unicode[STIX]{x1D6FC}}$ diffeomorphisms any weak mixing hyperbolic measure could be approximated by Bernoulli measures. This also holds true for $C^{1}$ diffeomorphisms preserving a weak mixing hyperbolic measure with respect to which the Oseledets decomposition is dominated.

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The Dynamics of the Weil–Petersson Flow

Matheus

2017

Lecture Notes in Mathematics

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Introduction 1.1. Some words on the origin of these notes 1.2. An overview of the dynamics of WP flow 1.3. Ergodicity of WP flow: outline of proof 1.4. Rates of mixing of WP flow 1.5. Organization of the text 2. Moduli spaces of Riemann surfaces and the Weil-Petersson metric 2.1. Definition and examples of moduli spaces 2.2. Teichmüller metric 2.3. Teichmüller spaces and mapping class groups 2.4. Fenchel-Nielsen coordinates 2.5. Cotangent bundle to moduli spaces of Riemann surfaces 2.6. Integrable quadratic differentials 2.7. Teichmüller and Weil-Petersson metrics 2.8. Ergodicity of WP flow: outline of proof revisited 3. Geometry of the Weil-Petersson metric 3.1. Items (I) and (II) of Theorem 1.5 for WP metric 3.2. Item (III) of Theorem 1.5 for WP metric 3.3. Item (IV) of Theorem 1.5 for WP metric 3.4. Item (V) of Theorem 1.5 for WP metric 3.5. Item (VI) of Theorem 1.5 for WP flow 4. Decay of correlations for the Weil-Petersson geodesic flow 4.1. Rates of mixing of the WP flow on T 1 M g ,n I: Proof of Theorem 4.1 4.2. Rates of mixing of the WP flow on T 1 M g ,n II: Proof of Theorem 4.2 References 2010

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The $C^{1+\alpha }$ hypothesis in Pesin Theory revisited

Cited by 20 publications

References 14 publications

Local stable and unstable sets for positive entropy $C^1$ dynamical systems

Local stable and unstable sets for positive entropy $C^1$ dynamical systems

Approximation of Bernoulli measures for non-uniformly hyperbolic systems

The Dynamics of the Weil–Petersson Flow

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