Stability Theorems for Group Actions on Uniform Spaces

TL;DR

This paper extends stability concepts from homeomorphisms to finitely generated group actions on uniform spaces, establishing conditions under which such actions are topologically stable, including measure-theoretic variants.

Contribution

It introduces measure-based stability notions for group actions on uniform spaces and proves stability results under these new conditions.

Findings

Expansive actions with shadowing or persistence are topologically stable.

Introduces measure-theoretic concepts like $$-expansivity and $$-stability.

Shows that measure-theoretic stability results parallel classical stability results.

Abstract

We extend the notions of topological stability, shadowing and persistence from homeomorphisms to finitely generated group actions on uniform spaces and prove that an expansive action with either shadowing or persistence is topologically stable. Using the concept of null set of a Borel measure $\mu$, we introduce the notions of $\mu$-expansivity, $\mu$-topological stability, $\mu$-shadowing and $\mu$-persistence for finitely generated group actions on uniform spaces and show that a $\mu$-expansive action with either $\mu$-shadowing or $\mu$-persistence is $\mu$-topologically stable.