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Metric-Independent Expansiveness

Alfonso Artigue, Luis Ferrari

Abstract

In this article we introduce and study a natural form of expansivity, that we call \textit{metric-independent expansiveness}, for group actions on metrizable spaces. This notion means \textit{expansive with respect to every compatible metric}. For actions on locally compact $σ$-compact metric spaces, we show that this property admits a purely topological characterization: it is equivalent to what we call \textit{cocompact expansivity} and to the existence of an expansive extension to the one-point compactification. We apply this characterization to ordinal spaces and to totally bounded spaces, obtaining criteria and examples that distinguish expansive actions from genuinely metric-independent ones. A central theme in these applications is that metric-independent expansiveness can be recovered from expansive compact dynamics when the boundary of the compactification is dynamically isolated. Finally, we introduce the notion of Cauchy expansiveness and prove that every uniformly continuous Cauchy expansive action extends uniquely to an expansive action on the completion.

Metric-Independent Expansiveness

Abstract

In this article we introduce and study a natural form of expansivity, that we call \textit{metric-independent expansiveness}, for group actions on metrizable spaces. This notion means \textit{expansive with respect to every compatible metric}. For actions on locally compact -compact metric spaces, we show that this property admits a purely topological characterization: it is equivalent to what we call \textit{cocompact expansivity} and to the existence of an expansive extension to the one-point compactification. We apply this characterization to ordinal spaces and to totally bounded spaces, obtaining criteria and examples that distinguish expansive actions from genuinely metric-independent ones. A central theme in these applications is that metric-independent expansiveness can be recovered from expansive compact dynamics when the boundary of the compactification is dynamically isolated. Finally, we introduce the notion of Cauchy expansiveness and prove that every uniformly continuous Cauchy expansive action extends uniquely to an expansive action on the completion.
Paper Structure (9 sections, 18 theorems, 45 equations)

This paper contains 9 sections, 18 theorems, 45 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Cocompact expansivity
  4. Preservation of Cocompact Expansivity under restrictions to subgroups
  5. Preservation of Cocompact Expansivity under Coproducts
  6. expansively extendible at a point
  7. Metric-Independent Expansivity in Ordinal Spaces
  8. Metric-Independent Expansivity in Totally Bounded Spaces
  9. Metric-Independent Expansivity and Completion of Metric Spaces

Key Result

Proposition 2.5

MIE is preserved under the restriction to a closed invariant subset.

Theorems & Definitions (59)

  • Definition 2.1
  • Example 2.2
  • Example 2.3
  • Remark 2.4
  • Proposition 2.5
  • proof
  • Definition 3.1
  • Remark 3.2
  • Proposition 3.3: Inheritance to syndetic subgroups
  • proof
  • ...and 49 more