A note on multivariate diam mean equicontinuity and frequent stability

Abstract

Let $(X,G)$ be a topological dynamical system, given by the action of a is a countable discrete infinite group on a compact metric space $X$. We prove that if $(X,G)$ is minimal, then it is either diam-mean $m$-equicontinuious or diam-mean $m$-sensitive. Similarly, $(X,G)$ is either frequently $m$-stable or strongly $m$-spreading. Further, when $G$ is abelian (or, more generally, virtually nilpotent), then the following statements are equivalent: $\bullet$ $(X,G)$ is a regular $m$-to-one extension of its maximal equicontinuous factor; $\bullet$ $(X,G)$ is diam-mean $(m+1)$-equicontinuious, and not diam mean $m$-equicontinuious; $\bullet$ $(X,G)$ is not diam-mean $(m+1)$-sensitive, but diam mean $m$-sensitive; $\bullet$ $(X,G)$ has an essential weakly mean sensitive $m$-tuple but no essential weakly mean sensitive $(m+1)$-tuple. This provides a {\em \enquote*{local}} characterisation of $m$-regularity and mean $m$-sensitivity vial weakly mean sensitive tuples. The same result holds when $G$ is amenable and $(X,G)$ satisfies the local Bronstein condition.