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Let $G$ be a countable infinite discrete amenable group.It should be notedthat 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 thecompact 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 outthat $\widetilde{\pi}$ is open if and only if $\pi$ is open. When $Y$ is fullysupported, it is shown that $\pi$ has relative uniformly positive entropy ifand only if $\widetilde{\pi}$ has relative uniformly positive entropy.

Relative uniformly positive entropy of induced amenable group actions