Mean Diameter, Regularity and Diam-Mean Equicontinuity

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.