Relative uniformly positive entropy of induced amenable group actions

TL;DR

This paper investigates the relationship between a $G$-system and its induced system on probability measures, establishing conditions under which properties like openness and relative uniformly positive entropy are preserved.

Contribution

It demonstrates that the openness and relative uniformly positive entropy of factor maps are equivalent between a $G$-system and its induced measure system for amenable groups.

Findings

Openness of the induced map $ ilde{ ho}$ is equivalent to the openness of $ ho$.

Relative uniformly positive entropy is preserved between the system and its measure-induced system.

The results apply to systems where the base space is fully supported.

Abstract

Let $G$ be a countable infinite discrete amenable group.It should be noted that a $G$-system $(X,G)$ naturally induces a $G$-system $(\mathcal{M}(X),G)$, where $\mathcal{M}(X)$ denotes the space of Borel probability measures on the compact metric space $X$ endowed with the weak*-topology. A factor map $\pi\colon (X,G)\to(Y,G)$ between two $G$-systems induces a factor map $\widetilde{\pi}\colon(\mathcal{M}(X),G)\to(\mathcal{M}(Y),G)$. It turns out that $\widetilde{\pi}$ is open if and only if $\pi$ is open. When $Y$ is fully supported, it is shown that $\pi$ has relative uniformly positive entropy if and only if $\widetilde{\pi}$ has relative uniformly positive entropy.