Topological dynamics of Polish group extensions

TL;DR

This paper investigates how the topological dynamics of a Polish group extension relate to the dynamics of its constituent groups, establishing conditions under which properties like metrizability and unique ergodicity are preserved.

Contribution

It proves that metrizability and unique ergodicity of universal minimal flows are preserved in certain Polish group extensions, providing new insights into their dynamical structure.

Findings

Metrizability of $M(H)$ and $M(K)$ implies metrizability of $M(G)$.

Unique ergodicity of $H$ and $K$ implies unique ergodicity of $G$.

Examples illustrating these dynamical phenomena.

Abstract

We consider a short exact sequence $1\to H\to G\to K\to 1$ of Polish groups and consider what can be deduced about the dynamics of $G$ given information about the dynamics of $H$ and $K$. We prove that if the respective universal minimal flows $M(H)$ and $M(K)$ are metrizable, then so is $M(G)$. Furthermore, we show that if $M(H)$ and $M(K)$ are metrizable and both $H$ and $K$ are uniquely ergodic, then so is $G$. We then discuss several examples of these phenomena