Relative uniformly positive entropy of induced amenable group actions

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 $π\colon (X,G)\to(Y,G)$ between two $G$-systems induces a factor map $\widetildeπ\colon(\mathcal{M}(X),G)\to(\mathcal{M}(Y),G)$. It turns out that $\widetildeπ$ is open if and only if $π$ is open. When $Y$ is fully supported, it is shown that $π$ has relative uniformly positive entropy if and only if $\widetildeπ$ has relative uniformly positive entropy.