Polish groups with metrizable universal minimal flows

TL;DR

This paper characterizes the structure of universal minimal flows for certain Polish groups with metrizable flows containing a G_delta orbit, linking topological dynamics with structural Ramsey theory and exploring proximal flows.

Contribution

It establishes a structural description of universal minimal flows with G_delta orbits and connects this to structural Ramsey theory, also analyzing proximal flows in examples.

Findings

Universal minimal flows with G_delta orbits are isomorphic to the completion of homogeneous spaces.

The results connect topological dynamics with structural Ramsey theory.

Concrete representations of universal minimal proximal flows are provided in examples.

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.