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.

在本文中，我们介绍了拓扑稳定性，遮蔽和弱遮蔽属性的概念，这些概念是非紧凑度量空间上某些有限生成的组的作用，它们是动态属性，并且与紧凑度量空间上的经典定义等效。此外，我们将Walter的稳定性定理扩展为对局部紧度量空间上的动作进行分组。我们表明，行动\：（\ varphi \ mathbb {F} _ {2} \乘M \至M \）自由基的˚F ₂ = <一个，b >上紧凑歧管中号与d我中号（中号）≥2当且仅当它具有较弱的阴影属性时，它才具有阴影属性。这意味着如果\（\ varphi：\ mathbb {F} _ {2} \ timesM \ to M \）在拓扑上是稳定的，因此它具有阴影属性。最后，我们引入一个度量\（\ mu \ in {\ mathscr {M}}（X）\）与连续动作φ的弱阴影特性兼容：G × X → X，用\（\ mu \ in {\ mathscr {M}}（X，\ varphi）\）中。我们表明，如果φ可以近似为μ -transitive动作一些\（在\亩\ {\ mathscr {M}}（X，\ varphi）\） ，然后Ω（φ）= X。我们举一个例子来说明它可能是\（{\ mathscr {M}}（X，\ varphi）\ neq \ emptyset \）而φ不具有弱跟踪性，但我们表明，如果有一个严格测量\（\亩\在{\ mathscr {M}}（X，\ varphi）\） ，然后连续动作φ的ģ上紧致度量空间X具有较弱的阴影属性。另外，我们证明了，如果G是一个有限生成的阿贝尔群，并且\（{\ mathscr {M}}（X，\ varphi）\ neq \ emptyset \），则\（{\ mathscr {M}}（X ，\ varphi）\）具有φ-不变测度。