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N-expansive homeomorphisms with the shadowing property

Bernardo Carvalho, Welington Cordeiro

Abstract

We discuss the dynamics of $n$-expansive homeomorphisms with the shadowing property defined on compact metric spaces. For every $n\in\mathbb{N}$, we exhibit an $n$-expansive homeomorphism, which is not $(n-1)$-expansive, has the shadowing property and admits an infinite number of chain-recurrent classes. We discuss some properties of the local stable (unstable) sets of $n$-expansive homeomorphisms with the shadowing property and use them to prove that some types of the limit shadowing property are present. This deals some direction to the problem of non-existence of topologically mixing $n$-expansive homeomorphisms that are not expansive.

N-expansive homeomorphisms with the shadowing property

Abstract

We discuss the dynamics of -expansive homeomorphisms with the shadowing property defined on compact metric spaces. For every , we exhibit an -expansive homeomorphism, which is not -expansive, has the shadowing property and admits an infinite number of chain-recurrent classes. We discuss some properties of the local stable (unstable) sets of -expansive homeomorphisms with the shadowing property and use them to prove that some types of the limit shadowing property are present. This deals some direction to the problem of non-existence of topologically mixing -expansive homeomorphisms that are not expansive.

Paper Structure

This paper contains 5 sections, 9 theorems, 88 equations.

Table of Contents

  1. Introduction and Statement of Results
  2. Definitions
  3. Proof of Theorem \ref{['Example']}
  4. Local stable sets
  5. Proof of Theorem \ref{['B']}

Key Result

Theorem A

For every $n\in\mathbb{N}$, there is an $n$-expansive homeomorphism, defined in a compact metric space, that is not $(n-1)$-expansive, has the shadowing property and admits an infinite number of chain recurrent classes.

Theorems & Definitions (30)

  • Definition 1.1
  • Theorem A
  • Theorem B
  • Theorem C
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • ...and 20 more