Polish groups with metrizable universal minimal flows

Abstract

We prove that if the universal minimal flow of a Polish group $G$ is metrizable and contains a $G_{\delta }$ orbit $G\cdot x_0$, then it is isomorphic to the completion of the homogeneous space $G/G_{x_0}$ and show how this result translates naturally in terms of structural Ramsey theory. We also investigate universal minimal proximal flows and describe concrete representations of them in a number of examples.

arXiv comment15 pages; post-refereed version