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arXiv · 2101.05935

Weak mean equicontinuity for a countable discrete amenable group action

Leiye Xu·Liqi Zheng
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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 $\{\ol{Orb(x)}, G\}$.

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Leiye Xu, Liqi Zheng. 2021-01-15. Weak mean equicontinuity for a countable discrete amenable group action. https://arxiv.org/abs/2101.05935

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