Dynamics and structure of groups of homeomorphisms of scattered spaces

TL;DR

This paper investigates the topological and dynamical properties of groups of homeomorphisms on scattered spaces, establishing key structural features and classifying their subgroups and flows, including the homeomorphism groups of compact ordinal spaces.

Contribution

It provides a comprehensive analysis of the structure, dynamics, and classification of homeomorphism groups of scattered spaces, including new results on Roelcke-precompactness and universal minimal flows.

Findings

Established Roelcke-precompactness and amenability for large classes of homeomorphism groups.

Classified all closed normal subgroups of these groups.

Computed the universal minimal flow and classified homeomorphism groups of compact ordinal spaces.

Abstract

We study the topological structure and the topological dynamics of groups of homeomorphisms of scattered spaces. For a large class of them (including the homeomorphism group of any ordinal space or of any locally compact scattered space), we establish Roelcke-precompactness and amenability, classify all closed normal subgroups and compute the universal minimal flow. As a by-product, we classify up to isomorphism the homeomorphism groups of compact ordinal spaces.