Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Some consequences of shadowing and entropy-expansiveness

Noriaki Kawaguchi

TL;DR

This work analyzes how shadowing and $h$-expansiveness constrain uniformly rigid behavior on compact metric spaces. It proves that any non-empty uniformly rigid subset under such a map $f$ must be zero-dimensional, which in turn forces the periodic-point set $Per(f)$ to be zero-dimensional when non-empty. The key method is a contradiction argument built from $\,\delta$-pseudo orbits and a symbolic shift factor, leading to a violation of $h$-expansiveness unless the uniformly rigid set is totally disconnected. The paper clarifies the interplay between rigidity, shadowing, and entropy-like constraints, and it provides several illustrative examples (Cantor-set identities, odometers, and shifts) highlighting the scope and limits of the results.

Abstract

We show that if a continuous self-map of a compact metric space is h-expansive and satisfies the shadowing property, then every non-empty uniformly rigid subset is zero-dimensional, and hence the set of periodic points is also zero-dimensional if it is not empty.

Some consequences of shadowing and entropy-expansiveness

TL;DR

This work analyzes how shadowing and -expansiveness constrain uniformly rigid behavior on compact metric spaces. It proves that any non-empty uniformly rigid subset under such a map must be zero-dimensional, which in turn forces the periodic-point set to be zero-dimensional when non-empty. The key method is a contradiction argument built from -pseudo orbits and a symbolic shift factor, leading to a violation of -expansiveness unless the uniformly rigid set is totally disconnected. The paper clarifies the interplay between rigidity, shadowing, and entropy-like constraints, and it provides several illustrative examples (Cantor-set identities, odometers, and shifts) highlighting the scope and limits of the results.

Abstract

We show that if a continuous self-map of a compact metric space is h-expansive and satisfies the shadowing property, then every non-empty uniformly rigid subset is zero-dimensional, and hence the set of periodic points is also zero-dimensional if it is not empty.

Paper Structure

This paper contains 3 sections, 8 theorems, 51 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem 1.1
  3. Examples

Key Result

Theorem 1.1

If a continuous map $f\colon X\to X$ is h-expansive and has the shadowing property, then every non-empty uniformly rigid subset $S$ of $X$ is zero-dimensional.

Theorems & Definitions (27)

  • Definition 1.1
  • Definition 1.2
  • Remark 1.1
  • Definition 1.3
  • Remark 1.2
  • Definition 1.4
  • Remark 1.3
  • Theorem 1.1
  • Lemma 1.1
  • Theorem 1.2
  • ...and 17 more