Topological stability and pseudo-orbit tracing property of group actions

TL;DR

This paper extends topological stability concepts to group actions on compact metric spaces, proving stability under certain conditions and characterizing subshifts of finite type via pseudo-orbit tracing.

Contribution

It generalizes topological stability to group actions and establishes conditions for stability and pseudo-orbit tracing in this broader context.

Findings

Group actions with expansiveness and pseudo-orbit tracing are topologically stable.

Provides a class of group actions exhibiting stability or pseudo-orbit tracing.

Characterizes subshifts of finite type over finitely generated groups using pseudo-orbit tracing.

Abstract

In this note we extend the concept of topological stability from homeomorphisms to group actions on compact metric spaces, and prove that if an action of a finitely generated group is expansive and has the pseudo-orbit tracing property then it is topologically stable. This represents a group action version of the Walter's stability theorem. Moreover we give a class of group actions with topological stability or pseudo-orbit tracing property. On the other hand, we also provide a characterization of subshifts of finite type over finitely generated groups in term of pseudo-orbit tracing property.