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Non-hyperbolic ergodic measures and horseshoes in partially hyperbolic homoclinic classes

Dawei Yang, Jinhua Zhang

Abstract

We study a rich family of robustly non-hyperbolic transitive diffeomorphisms and we show that each ergodic measure is approached by hyperbolic sets in weak$*$-topology and in entropy. For hyperbolic ergodic measures, it is a classical result of A. Katok. The novelty here is to deal with non-hyperbolic ergodic measures.

Non-hyperbolic ergodic measures and horseshoes in partially hyperbolic homoclinic classes

Abstract

We study a rich family of robustly non-hyperbolic transitive diffeomorphisms and we show that each ergodic measure is approached by hyperbolic sets in weak-topology and in entropy. For hyperbolic ergodic measures, it is a classical result of A. Katok. The novelty here is to deal with non-hyperbolic ergodic measures.

Paper Structure

This paper contains 13 sections, 19 theorems, 54 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Entropy for horseshoes
  4. Partial hyperbolicity
  5. A shadowing lemma
  6. Pliss lemma
  7. Blender
  8. Metric entropy defined by A. Katok
  9. Periodic orbits near non-hyperbolic ergodic measures
  10. A sufficient condition for horseshoes with large entropy
  11. Horseshoes derived from separated sets
  12. Pesin blocks and periodic points close to non-hyperbolic ergodic measures
  13. Non-hyperbolic ergodic measures approached by hyperbolic basic sets

Key Result

Theorem A

There exists a $C^1$ open and dense subset $\mathcal{V}(M)$ of $\mathcal{U}(M)$ such that for any $f\in\mathcal{V}(M)$, each $f$-ergodic measure $\mu$ is approached by hyperbolic sets in weak$*$-topology and in entropy, that is, for any $\varepsilon>0$, there exists a hyperbolic set $\Lambda_{\varep Moreover, if $\mu$ is non-hyperbolic, $\mu$ can be approxiamted by hyperbolic sets of different ind

Figures (1)

  • Figure 1: The itinerary of $z$

Theorems & Definitions (26)

  • Theorem A
  • Corollary B
  • Remark \oldthetheorem
  • Corollary \oldthetheorem
  • Theorem \oldthetheorem
  • Lemma \oldthetheorem
  • Corollary \oldthetheorem
  • Lemma \oldthetheorem
  • Lemma \oldthetheorem
  • Lemma \oldthetheorem
  • ...and 16 more