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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.

Topological stability and pseudo-orbit tracing property of group actions