Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Continuum-wise hyperbolicity and periodic points

Bernardo Carvalho, Piotr Oprocha, Elias Rego

Abstract

We prove that cw-hyperbolic homeomorphisms with jointly continuous stable/unstable holonomies satisfy the periodic shadowing property and, if they are topologically mixing, the periodic specification property. We discuss difficulties to adapt Bowen's techniques to obtain a measure of maximal entropy for cw-hyperbolic homeomorphisms, exhibit the unique measure of maximal entropy for Walter's pseudo-Anosov diffeomorphism of $\mathbb{S}^2$, and prove it can be obtained, as in the expansive case, as the weak* limit of an average of Dirac measures on periodic orbits. As an application, we exhibit the unique measure of maximal entropy for the homeomorphism on the Sierpiński Carpet defined in [12], which does not satisfy the specification property.

Continuum-wise hyperbolicity and periodic points

Abstract

We prove that cw-hyperbolic homeomorphisms with jointly continuous stable/unstable holonomies satisfy the periodic shadowing property and, if they are topologically mixing, the periodic specification property. We discuss difficulties to adapt Bowen's techniques to obtain a measure of maximal entropy for cw-hyperbolic homeomorphisms, exhibit the unique measure of maximal entropy for Walter's pseudo-Anosov diffeomorphism of , and prove it can be obtained, as in the expansive case, as the weak* limit of an average of Dirac measures on periodic orbits. As an application, we exhibit the unique measure of maximal entropy for the homeomorphism on the Sierpiński Carpet defined in [12], which does not satisfy the specification property.

Paper Structure

This paper contains 5 sections, 11 theorems, 99 equations, 1 table.

Table of Contents

  1. Introduction
  2. Periodic shadowing and periodic specification
  3. Walter's pseudo-Anosov diffeomorphism of $\mathbb{S}^2$
  4. Measure of Maximal Entropy on the Sierpiński Carpet
  5. Further remarks and questions

Key Result

Theorem 2.4

If $f\colon X\to X$ is a cw-expansive homeomorphism of a compact metric space $X$, then there is a function $D\colon E\to\mathbb{R}$ satisfying the following conditions.

Theorems & Definitions (32)

  • Definition 1.1: cw-hyperbolicity
  • Definition 2.1: Shadowing
  • Definition 2.2
  • Definition 2.3: Local stable and local unstable holonomies
  • Theorem 2.4: Hyperbolic $cw$-metric-ACCV3
  • Definition 2.5
  • Definition 2.6: Jointly pseudo-isometric stable/unstable holonomies
  • Theorem 2.7
  • proof
  • Remark 2.8
  • ...and 22 more