Relative uniformly positive entropy of induced amenable group actions

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