Multivariate mean equicontinuity for finite-to-one topomorphic extensions

In this note, we generalise the concept of topo-isomorphic extensions and define finite topomorphic extensions as topological dynamical systems whose factor map to the maximal equicontinuous factor is measure-theoretically at most $m$-to-one for some $m\in\mathbb{N}$. We further define multivariate versions of mean equicontinuity, complementing the notion of multivariate mean sensitivity introduced by Li, Ye and Yu, and then show that any $m$-to-one topomorphic extension is mean $(m+1)$-equicontinuous. This falls in line with the well-known result, due to Downarowicz and Glasner, that strictly ergodic systems are isomorphic extensions if and only if they are mean equicontinuous. While in the multivariate case we can only conjecture that the converse direction also holds, the result provides an indication that multivariate equicontinuity properties are strongly related to finite extension structures. For minimal systems, an Auslander-Yorke type dichotomy between multivariate mean equicontinuity and sensitivity is shown as well.