Metric-Independent Expansiveness

TL;DR

This paper introduces and characterizes metric-independent expansiveness for group actions on metrizable spaces, providing topological criteria, examples, and extensions related to compactifications and Cauchy expansiveness.

Contribution

It defines metric-independent expansiveness, establishes its topological characterization, and explores its implications and extensions in various classes of spaces.

Findings

Metric-independent expansiveness is equivalent to cocompact expansivity on certain spaces.

Expansive actions can be extended to compactifications with isolated boundaries.

Cauchy expansive actions extend uniquely to the completion, preserving 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 $\sigma$-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.