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Sensitivity, local stable/unstable sets and shadowing

Mayara Antunes, Bernardo Carvalho, Margoth Tacuri

Abstract

In this paper we study local stable/unstable sets of sensitive homeomorphisms with the shadowing property defined on compact metric spaces. We prove that local stable/unstable sets always contain a compact and perfect subset of the space. As a corollary we generalize results in [6] and [11] proving that positively countably expansive homeomorphisms defined on compact metric spaces satisfying either transitivity and the shadowing property, or the L-shadowing property, can only be defined in countably spaces.

Sensitivity, local stable/unstable sets and shadowing

Abstract

In this paper we study local stable/unstable sets of sensitive homeomorphisms with the shadowing property defined on compact metric spaces. We prove that local stable/unstable sets always contain a compact and perfect subset of the space. As a corollary we generalize results in [6] and [11] proving that positively countably expansive homeomorphisms defined on compact metric spaces satisfying either transitivity and the shadowing property, or the L-shadowing property, can only be defined in countably spaces.

Paper Structure

This paper contains 3 sections, 7 theorems, 59 equations.

Table of Contents

  1. Introduction and the main result
  2. Positive cw-expansivity and shadowing
  3. Examples

Key Result

Theorem A

Let $f\colon X\to X$ be a homeomorphism of a compact metric space $X$ satisfying the shadowing property.

Theorems & Definitions (25)

  • Definition 1.1
  • Definition 1.2
  • Theorem A
  • proof
  • Definition 2.1
  • Theorem 2.2
  • proof
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • ...and 15 more