Shadowing and mixing on systems of countable group actions

TL;DR

This paper characterizes shadowing and mixing properties in dynamical systems with countable group actions, linking them to inverse limits of shifts of finite type and their structural conditions.

Contribution

It provides new characterizations of shadowing and mixing properties for systems with countable group actions, especially relating these properties to inverse limits and Mittag-Leffler conditions.

Findings

Shadowing property characterized by inverse limits of shifts of finite type.

Mixing and other properties characterized via inverse limits and Mittag-Leffler condition.

Results apply to both totally disconnected and metric space systems.

Abstract

Let $(X,G,\Phi)$ be a dynamical system, where $X$ is compact Hausdorff space, and $G$ is a countable discrete group. We investigate shadowing property and mixing between subshifts and general dynamical systems. For the shadowing property, fix some finite subset $S\subset G$. We prove that if $X$ is totally disconnected, then $\Phi$ has $S$-shadowing property if and only if $(X,G,\Phi)$ is conjugate to an inverse limit of a sequence of shifts of finite type which satisfies Mittag-Leffler condition. Also, suppose that $X$ is metric space (may be not totally disconnected), we prove that if $\Phi$ has $S$-shadowing property, then $(X,G,\Phi)$ is a factor of an inverse limit of a sequence of shifts of finite type by a factor map which almost lifts pseudo-orbit for $S$. On the other hand, let property $P$ be one of the following property: transitivity, minimal, totally transitivity, weakly mixing, mixing, and specification property. We prove that if $X$ is totally disconnected, then $\Phi$ has property $P$ if and only if $(X,G,\Phi)$ is conjugate to an inverse limit of an inverse system that consists of subshifts with property $P$ which satisfies Mittag-Leffler condition. Also, for the case of metric space (may be not totally disconnected), if property $P$ is not minimal or specification property, we prove that $\Phi$ has property $P$ if and only if $(X,G,\Phi)$ is a factor of an inverse limit of a sequence of subshifts with property $P$ which satisfies Mittag-Leffler condition.