Topological dynamics of Polish group extensions

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