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proximal and strongly proximal actions; probability measures
We prove that action of a semigroup $T$ on compact metric space $X$ by continuous selfmaps is strongly proximal if and only if $T$ action on ${\mathcal P}(X)$ is strongly proximal. As a consequence we prove that affine actions on certain compact convex subsets of finite-dimensional vector spaces are strongly proximal if and only if the action is proximal.
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