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 $\pi\colon (X,G)\to(Y,G)$ 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.