Multivariate Frequent Stability and Diam-Mean Equicontinuity

TL;DR

This paper introduces multivariate versions of frequent stability and diam-mean equicontinuity, characterizing systems with finite-to-one factor maps to the maximal equicontinuous factor using these properties.

Contribution

It defines and studies multivariate stability and equicontinuity, linking them to almost finite-to-one extensions of the maximal equicontinuous factor in dynamical systems.

Findings

Frequent m-stability characterizes almost m-to-one extensions.

Diam-mean m-equicontinuity corresponds to almost surely m-to-one extensions.

Results apply to systems with abelian acting groups.

Abstract

In this paper, we introduce and investigate multivariate versions of frequent stability and diam-mean equicontinuity. Given a natural number $m > 1$, we call those notions "frequent $m$-stability" and "diam-mean $m$-equicontinuity". We use these dynamical rigidity properties to characterise systems whose factor map to the maximal equicontinuous factor (MEF) is finite-to-one for a residual set, called "almost finite-to-one extensions", or a set of full measure, called "almost surely finite-to-one extensions". In the case of a $\sigma$-compact, locally compact, abelian acting group it is shown that frequently $(m+1)$-stable systems are equivalently characterised as almost $m$-to-one extensions of their MEF. Similarly, it is shown that a system is diam-mean $(m+1)$-equicontinuous if and only if it is an almost surely $m$-to-one extension of its MEF.