Persistent Shadowing For Actions Of Some Finitely Generated Groups and Related Measures
In this paper, $φ:G\times X\to X$ 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 $φ:G\times X\to X$ and study it via measure theory. Indeed, we introduce the notion of compatibility the Borel probability measure $μ$ with respect persistent shadowing property of $φ:G\times X\to X$ and denote it by $μ\in\mathcal{M}_{PSh}(X, φ)$. We show $μ\in\mathcal{M}_{PSh}(X, φ)$ if and only if $supp(μ)\subseteq PSh(φ)$, where $PSh(φ)$ is the set of all persistent shadowable points of $φ$. This implies that if every non-atomic Borel probability measure $μ$ is compatible with persistent shadowing property for $φ:G\times X\to X$, then $φ$ does have persistent shadowing property. We prove that $\overline{PSh(φ)}=PSh(φ)$ if and only if $\overline{\mathcal{M}_{PSh}(X, φ)}= \mathcal{M}_{PSh}(X, φ)$. Also, $μ(\overline{PSh(φ)})=1$ if and only if $μ\in\overline{\mathcal{M}_{PSh}(X, φ)}$. Finally, we show that $\overline{\mathcal{M}_{PSh}(X, φ)}=\mathcal{M}(X)$ if and only if $\overline{PSh(φ)}=X$. For study of persistent shadowing property, we introduce the notions of uniformly $α$-persistent point, uniformly $β$-persistent point and recall notions of shadowing property, $α$-persistent, $β$-persistent and we give some further results about them.