Weak mean equicontinuity for a countable discrete amenable group action

TL;DR

This paper investigates weak mean equicontinuity in countable discrete amenable group actions on compact spaces, establishing equivalences with mean equicontinuity and characterizations involving ergodic measures.

Contribution

It introduces the concept of weak mean equicontinuity for group actions and proves its equivalence to mean equicontinuity under certain conditions, providing new characterizations.

Findings

Weak mean equicontinuity is equivalent to mean equicontinuity for the product action.

When the system has full measure center or the group is abelian, weak mean equicontinuity characterizes uniquely ergodic points.

Continuity of the map to ergodic measures characterizes weak mean equicontinuity in specific cases.

Abstract

The weak mean equicontinuous properties for a countable discrete amenable group $G$ acting continuously on a compact metrizable space $X$ are studied. It is shown that the weak mean equicontinuity of $(X \times X,G)$ is equivalent to the mean equicontinuity of $(X,G)$. Moreover, when $(X,G)$ has full measure center or $G$ is abelian, it is shown that $(X,G)$ is weak mean equicontinuous if and only if all points in $X$ are uniquely ergodic points and the map $x \to \mu_x^G$ is continuous, where $\mu_x^G$ is the unique ergodic measure on $\{\ol{Orb(x)}, G\}$.