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

TL;DR

This paper introduces and analyzes the concept of persistent shadowing in group actions on compact metric spaces, linking it with measure theory and exploring conditions for its prevalence and properties.

Contribution

It defines persistent shadowing for group actions, relates it to measure-theoretic properties, and establishes conditions for its genericity and stability within the space of measures.

Findings

Persistent shadowing characterized via support of measures.

Conditions for measures to be compatible with persistent shadowing.

Equivalence between closure properties of shadowing points and measures.

Abstract

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.