Mean Diameter, Regularity and Diam-Mean Equicontinuity

TL;DR

This paper introduces the concept of diam-mean equicontinuity in group actions, characterizes regular factor maps using mean diameter, and establishes the existence of a maximal diam-mean equicontinuous factor for amenable group actions.

Contribution

It provides a new characterization of diam-mean equicontinuity and proves the existence of a maximal such factor in the context of amenable group actions.

Findings

Diam-mean equicontinuity characterized by regular extensions.

Existence of a maximal diam-mean equicontinuous factor.

Stability properties of regular factor maps established.

Abstract

In the context of (not necessarily minimal) actions, we consider the mean diameter and use it to characterize regular factor maps. Building on this characterization, we prove that an action is diam-mean equicontinuous if and only if it is a regular extension of its maximal equicontinuous factor. Furthermore, we establish the existence of a maximal diam-mean equicontinuous factor and discuss stability properties of regular factor maps. For this, we work in the context of actions of locally compact and $\sigma$-compact amenable groups.