Veech's Theorem of $G$ acting freely on $G^{\textrm{LUC}}$ and Structure Theorem of a.a. flows

TL;DR

This paper extends Veech's Theorem to broader classes of groups and explores the structure of almost automorphic flows, establishing their uniqueness and characterizations in topological dynamics.

Contribution

It proves Veech's Theorem for locally quasi-totally bounded groups and characterizes the structure and uniqueness of the universal almost automorphic flow.

Findings

Veech's Theorem holds for locally quasi-totally bounded groups.

The universal a.a. flow is the maximal almost 1-1 extension of the universal minimal a.p. flow.

Every endomorphism of Veech's hull flow induced by an a.a. function is almost 1-1.

Abstract

Veech's Theorem claims that if $G$ is a locally compact\,(LC) Hausdorff topological group, then it may act freely on $G^{\textrm{LUC}}$. We prove Veech's Theorem for $G$ being only locally quasi-totally bounded, not necessarily LC. And we show that the universal a.a. flow is the maximal almost 1-1 extension of the universal minimal a.p. flow and is unique up to almost 1-1 extensions. In particular, every endomorphism of Veech's hull flow induced by an a.a. function is almost 1-1; for $G=\mathbb{Z}$ or $\mathbb{R}$, $G$ acts freely on its canonical universal a.a. space. Finally, we characterize Bochner a.a. functions on a LC group $G$ in terms of Bohr a.a. function on $G$ (due to Veech 1965 for the special case that $G$ is abelian, LC, $\sigma$-compact, and first countable).