Weak Shadowing for Actions of Some Finitely Generated Groups on Non-compact Spaces and Related Measures

Abstract

In this paper, we introduce the notions of topological stability, shadowing, and weak shadowing properties for actions of some finitely generated groups on non-compact metric spaces which are dynamical properties and equivalent to the classical definitions in case of compact metric spaces. Also, we extend Walter’s stability theorem to group actions on locally compact metric spaces. We show that action $\varphi :\mathbb {F}_{2}\times M\to M$ of free group F₂ =< a, b > on compact manifold M with dim(M) ≥ 2 has shadowing property if and only if it has weak shadowing property. This implies that if $\varphi :\mathbb {F}_{2}\times M\to M$ is topologically stable, then it has shadowing property. Finally, we introduce a measure $\mu \in {\mathscr{M}}(X)$ that is compatible with the weak shadowing property for the continuous action φ : G × X → X, denoted by $\mu \in {\mathscr{M}}(X, \varphi )$. We show that if φ can be approximated by μ-transitive actions for some $\mu \in {\mathscr{M}}(X, \varphi )$, then Ω(φ) = X. We give an example to show that it may be ${\mathscr{M}}(X, \varphi )\neq \emptyset$ while φ does not have weak shadowing property but we show that if there is a strictly measure $\mu \in {\mathscr{M}}(X, \varphi )$, then continuous action φ of G on compact metric space X has weak shadowing property. Also, we show that if G is a finitely generated abelian group and ${\mathscr{M}}(X, \varphi )\neq \emptyset$, then closure of ${\mathscr{M}}(X, \varphi )$ has an φ-invariant measure.