Reflections on equicontinuity

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 the “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.