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SRB measures for mostly expanding partially hyperbolic diffeomorphisms via the variational approach

David Burguet, Dawei Yang

Abstract

By using the variational approach, we prove the existence of Sinai-Ruelle-Bowen measures for partially hyperbolic $\mathcal C^1$ diffeomorphisms with mostly expanding properties. The same conclusion holds true if one considers a dominated splitting $E\oplus F$, where $\dim E=1$ and $F$ is mostly expanding. When the diffeomorphisms are $\mathcal C^{1+α}$, we prove the basin covering property for both cases.

SRB measures for mostly expanding partially hyperbolic diffeomorphisms via the variational approach

Abstract

By using the variational approach, we prove the existence of Sinai-Ruelle-Bowen measures for partially hyperbolic diffeomorphisms with mostly expanding properties. The same conclusion holds true if one considers a dominated splitting , where and is mostly expanding. When the diffeomorphisms are , we prove the basin covering property for both cases.
Paper Structure (16 sections, 21 theorems, 113 equations)

This paper contains 16 sections, 21 theorems, 113 equations.

Table of Contents

  1. Introduction
  2. Construction of SRB measure
  3. Submanifolds tangent to a cone around $F$
  4. Pliss times
  5. Fø lner empirical measures
  6. Fø lner Gibbs Property
  7. Ergodic properties of the limit empirical measure $\mu$
  8. The basin
  9. Dynamical density points on Pliss times
  10. Initialization:
  11. Induction step:
  12. Proof of Corollary \ref{['corConeplus']}
  13. Finiteness
  14. Plaque family theorem and stable manifolds
  15. Bi-Pliss times
  16. ...and 1 more sections

Key Result

Theorem 1

Assume that $\Lambda=\bigcap_{n\in {\mathbb N}}f^nU$ is an attractor of a $\mathcal{C}^{1+\alpha}$ diffeomorphism $f$ with a dominated splitting $T_\Lambda M=E\oplus F$ with $E$ being uniformly contracting. Then Lebesgue a.e. $x\in U$ with lies in the basin of an ergodic hyperbolic SRB measure. Moreover for any $a>0$, the set $\left\{x\in U, \ \overline{m}_F(x,f)>a\right\}$ may be covered Lebes

Theorems & Definitions (36)

  • Theorem : ABV00ADLP
  • Theorem 1
  • Corollary 2
  • Corollary 3
  • Theorem 4
  • Theorem 5
  • Lemma 2.1
  • proof
  • Lemma 2.2: Lemma 3.1 in ABV00
  • Lemma 2.3: Lemma 4.2 in AlP08
  • ...and 26 more