Persistent Shadowing For Actions Of Some Finitely Generated Groups and Related Measures

In this paper, $\varphi:G\times X\to X$ is a continuous action of finitely generated group $G$ on compact metric space $(X, d)$ without isolated point. We introduce the notion of persistent shadowing property for $\varphi:G\times X\to X$ and study it via measure theory. Indeed, we introduce the notion of compatibility the Borel probability measure $\mu$ with respect persistent shadowing property of $\varphi:G\times X\to X$ and denote it by $\mu\in\mathcal{M}_{PSh}(X, \varphi)$. We show $\mu\in\mathcal{M}_{PSh}(X, \varphi)$ if and only if $supp(\mu)\subseteq PSh(\varphi)$, where $PSh(\varphi)$ is the set of all persistent shadowable points of $\varphi$. This implies that if every non-atomic Borel probability measure $\mu$ is compatible with persistent shadowing property for $\varphi:G\times X\to X$, then $\varphi$ does have persistent shadowing property. We prove that $\overline{PSh(\varphi)}=PSh(\varphi)$ if and only if $\overline{\mathcal{M}_{PSh}(X, \varphi)}= \mathcal{M}_{PSh}(X, \varphi)$. Also, $\mu(\overline{PSh(\varphi)})=1$ if and only if $\mu\in\overline{\mathcal{M}_{PSh}(X, \varphi)}$. Finally, we show that $\overline{\mathcal{M}_{PSh}(X, \varphi)}=\mathcal{M}(X)$ if and only if $\overline{PSh(\varphi)}=X$. For study of persistent shadowing property, we introduce the notions of uniformly $\alpha$-persistent point, uniformly $\beta$-persistent point and recall notions of shadowing property, $\alpha$-persistent, $\beta$-persistent and we give some further results about them.