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Expansiveness for flows on noncompact spaces

Y. Yang, C. A. Morales

Abstract

We introduce the concept of topological expansive flow. We prove that this concept is invariant by topological conjugacy and reduces to expansivity in the compact case. We characterize tiopological expansive flows as rescaling expansive flows for which the singularities are isolated points of the space. Finally, we prove that the growth rate of the periodic orbits of a topological expansive flow which is dynamically isolated at infinity is controlled by an entropy-like invariant. This extends Bowen-Walters inequality for expansive flows on compact spaces.

Expansiveness for flows on noncompact spaces

Abstract

We introduce the concept of topological expansive flow. We prove that this concept is invariant by topological conjugacy and reduces to expansivity in the compact case. We characterize tiopological expansive flows as rescaling expansive flows for which the singularities are isolated points of the space. Finally, we prove that the growth rate of the periodic orbits of a topological expansive flow which is dynamically isolated at infinity is controlled by an entropy-like invariant. This extends Bowen-Walters inequality for expansive flows on compact spaces.
Paper Structure (3 sections, 14 theorems, 78 equations)

This paper contains 3 sections, 14 theorems, 78 equations.

Table of Contents

  1. Introduction
  2. Preliminary lemmas
  3. Proof of the theorems

Key Result

Theorem 1

Topological expansivity is invariant under topological conjugacy and reduces to expansivity on compact metric spaces. More precisely, a flow of a compact metric space is topological expansive if and only if it is expansive.

Theorems & Definitions (36)

  • Example 1.1
  • Definition 1.1
  • Example 1.2
  • Definition 1.2
  • Example 1.3
  • Theorem 1
  • Example 1.4
  • Example 1.5
  • proof
  • Example 1.6
  • ...and 26 more