Universal minimal flows of generalized Wa\.zewski dendrites

TL;DR

This paper investigates the universal minimal flows of homeomorphism groups of generalized Ważewski dendrites, providing explicit descriptions for finite cases and demonstrating non-metrizability for infinite cases, thus advancing understanding of topological group dynamics.

Contribution

It explicitly computes universal minimal flows for finite generalized Ważewski dendrites and shows non-metrizability in the infinite case, answering a previously open question.

Findings

Universal minimal flow is metrizable for finite P.

Explicit computation of the universal minimal flow for finite P.

Universal minimal flow is non-metrizable for infinite P.

Abstract

We study universal minimal flows of the homeomorphism groups of generalized Wa\.zewski dendrites $W_P$, $P\subset\{3,4,\ldots,\omega\}$. If $P$ is finite, we prove that the universal minimal flow of of the homeomorphism group $H(W_P)$ is metrizable and we compute it explicitly. This answers a question of B. Duchesne. If $P$ is infinite, we show that the universal minimal flow of $H(W_P)$ is not metrizable. This provides examples of topological groups which are Roelcke precompact and have a non-metrizable universal minimal flow with a comeager orbit.