Reflections on equicontinuity

TL;DR

This paper explores various equivalent conditions for equicontinuity in transitive compact Hausdorff flows with general group actions, including regional equicontinuity and almost automorphy, and extends Veech's results to non-Abelian groups.

Contribution

It introduces new equivalences for equicontinuity and generalizes Veech's results to broader group actions within dynamical systems.

Findings

Equivalence of multiple conditions to equicontinuity in general group actions.

Extension of Veech's results to non-Abelian groups.

Analysis of relations like regionally proximal and Veech's relations in this context.

Abstract

We study different conditions which turn out to be equivalent to equicontinuity for a transitive compact Hausdorff flow with a general group action. Among them are a notion of "regional" equicontinuity, also known as "Furstenberg" condition, and the condition that every point of the phase space is almost automorphic. Then we study relations on the phase space arising from dynamical properties, among them the regionally proximal relation and two relations introduced by Veech. We generalize Veech's results for minimal actions of non-Abelian groups preserving a probability measure with respect to the regionally proximal relation. We provide proofs in the framework of dynamical systems rather than harmonic analysis as given by Veech.