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Keywords:
proximal and strongly proximal actions; probability measures
proximal and strongly proximal actions; probability measures
Summary:
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.
References:
[B] Billingley P.: Convergence of Probability Measures. John Willey and Sons, New York-Toronto, 1968. MR 0233396
[G] Glasner S.: Proximal flows on Lie groups. Israel Journal of Mathematics 45 (1983), 97–99. MR 0719114
[P] Parthasarathy K. R.: Probability Measures on Metric Spaces. Academic Press, New York-London, 1967. MR 0226684 | Zbl 0153.19101
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