Projectivity in topological dynamics

TL;DR

This paper investigates projectivity in topological dynamics for Polish groups, introduces proximally irreducible extensions, and characterizes various forms of amenability through dynamical properties of Samuel compactifications.

Contribution

It provides a new characterization of amenability and extreme amenability via dynamical irreducibility properties and answers an open question about the structure of universal minimal proximal flows.

Findings

Characterization of extreme and strong amenability using Samuel compactification properties

Introduction of proximally irreducible extensions between affine G-flows

Structure theorem for the metrizability and orbit properties of the universal minimal proximal flow

Abstract

We study projectivity in the category of $G$-flows and affine $G$-flows for Polish groups $G$. We also introduce the notion of \emph{proximally irreducible} extensions between affine $G$-flows. Using this we provide a characterization of extreme amenability, strong amenability, and amenability for closed subgroups $H \leq G$ in terms of certain ``dynamical irreducibility'' properties of the Samuel compactification of $G/H$. We then apply this to answer an open question of Zucker by proving a structure theorem for when the universal minimal proximal flow of $G$ is metrizable or contains a comeager orbit.