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Entropy properties of mostly expanding partially hyperbolic diffeomorphisms

Jinhua Zhang

Abstract

The statistical properties of mostly expanding partially hyperbolic diffeomorphisms have been substantially studied. In this paper, we would like to address the entropy properties of mostly expanding partially hyperbolic diffeomorphisms. We prove that for mostly expanding partially hyperbolic diffeomorphisms with minimal strong stable foliation and one-dimensional center bundle, there exists a $C^1$-open neighborhood of them, in which the topological entropy varies continuously and the intermediate entropy property holds. To prove that, we show that each non-hyperbolic ergodic measure is approached by horseshoes in entropy and in weak$*$-topology.

Entropy properties of mostly expanding partially hyperbolic diffeomorphisms

Abstract

The statistical properties of mostly expanding partially hyperbolic diffeomorphisms have been substantially studied. In this paper, we would like to address the entropy properties of mostly expanding partially hyperbolic diffeomorphisms. We prove that for mostly expanding partially hyperbolic diffeomorphisms with minimal strong stable foliation and one-dimensional center bundle, there exists a -open neighborhood of them, in which the topological entropy varies continuously and the intermediate entropy property holds. To prove that, we show that each non-hyperbolic ergodic measure is approached by horseshoes in entropy and in weak-topology.
Paper Structure (13 sections, 25 theorems, 106 equations)

This paper contains 13 sections, 25 theorems, 106 equations.

Table of Contents

  1. Introduction
  2. Statements of our main results
  3. Preliminaries
  4. Topological entropy and metric entropy
  5. Liao's shadowing lemma and Pliss lemma
  6. Plaque families and uniform size of invariant manifolds
  7. Constructions of horseshoes from hyperbolic periodic orbits
  8. Unstable entropy and entropy formulas
  9. Mostly expanding partially hyperbolic diffeomorphisms
  10. Existence of periodic orbits with prescribed behavior
  11. Generating orbit segments with weak hyperbolicity
  12. Proof of Theorem \ref{['thm.key-theorem']}
  13. Approximation of ergodic measures by horseshoes: Proofs of our main results

Key Result

Theorem 1

Let $f\in\operatorname{Diff}^{1}(M)$ be a partially hyperbolic diffeomorphism with $\operatorname{dim}(E^c)=1$. Assume that Then there exists a $C^1$-open neighborhood $\mathcal{U}\subset \operatorname{Diff}^1(M)$ of $f$ such that

Theorems & Definitions (39)

  • Theorem 1
  • Remark 1.1
  • Corollary 2
  • Remark 1.2
  • Theorem 3
  • Remark 1.3
  • Theorem 4
  • Remark 1.4
  • Remark 2.1
  • Proposition 2.2: Proposition 2 in B and Section 11 in Pe
  • ...and 29 more