A note on mean equicontinuity

TL;DR

This paper extends the understanding of mean equicontinuity in topological dynamical systems, showing equivalences among various forms and introducing a relation that characterizes the maximal mean equicontinuous factor.

Contribution

It generalizes previous results on mean equicontinuity from minimal systems to all topological dynamical systems and introduces a new relation for maximal mean equicontinuous factors.

Findings

Mean equicontinuity is equivalent to equicontinuity in the mean.

A system is Banach (or Weyl) mean equicontinuous if and only if its regionally proximal relation equals its Banach proximal relation.

A new relation induces the maximal mean equicontinuous factor for any system.

Abstract

In this note, it is shown that several results concerning mean equicontinuity proved before for minimal systems are actually held for general topological dynamical systems. Particularly, it turns out that a dynamical system is mean equicontinuous if and only if it is equicontinuous in the mean if and only if it is Banach (or Weyl) mean equicontinuous if and only if its regionally proximal relation is equal to the Banach proximal relation. Meanwhile, a relation is introduced such that the smallest closed invariant equivalence relation containing this relation induces the maximal mean equicontinuous factor for any system.