A note on multivariate diam mean equicontinuity and frequent stability

TL;DR

This paper investigates the properties of multivariate diam-mean equicontinuity and stability in minimal topological dynamical systems, establishing equivalences and characterizations especially for abelian or virtually nilpotent groups.

Contribution

It provides new characterizations of regularity and mean sensitivity in dynamical systems using weakly mean sensitive tuples, especially for abelian and amenable groups.

Findings

Dichotomy between diam-mean m-equicontinuity and diam-mean m-sensitivity in minimal systems.

Equivalence of several conditions for systems with abelian or virtually nilpotent groups.

Characterization of m-regularity via weakly mean sensitive tuples.

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.