Multivariate Frequent Stability and Diam-Mean Equicontinuity
Lino Haupt
TL;DR
The paper develops multivariate extensions of weak equicontinuity notions for actions of a σ-compact, locally compact, abelian group on a compact metric space, introducing the multivariate diameter via $D_m$ and $\mathsf{diam}_m$. It characterizes these properties through finite-to-one extensions of the maximal equicontinuous factor (MEF), distinguishing almost $m$-to-one and almost surely $m$-to-one fibers. The main results show that a minimal system is frequently $(m+1)$-stable precisely when the MEF fibers are almost $m$:1, and is diam-mean $(m+1)$-equicontinuous precisely when fibers are almost surely $m$:1, with proofs leveraging multivariate proximal relations, hyperspace dynamics, and ergodic theory. This work extends known one-to-one dichotomies to finite-to-one settings, offering a rigorous bridge between fiber structure and dynamical rigidity in the multivariate regime.
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 $σ$-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.