Conditional Topomorphic Degree and Multivariate Mean Equicontinuity for Minimal Amenable Group Actions

Abstract

Let $G$ be a countably infinite discrete amenable group acting minimally on a compact metric space $X$, and let $\pi :X\to X_{\mathrm{eq}}$ be the maximal equicontinuous factor map. We introduce the \emph{conditional topomorphic degree} $d:=\tdeg_G(X)\in\mathbb N\cup\{\infty\}$, which, when finite, is the least integer such that $\pi$ is an at most $d$-to-one topomorphic extension. We prove that, for every $r\ge 2$, Weyl mean $r$-equicontinuity, mean $r$-equicontinuity along some Følner sequence, and $d\le r-1$ are equivalent. For minimal $\mathbb{Z}$-systems, this settles a conjecture of Breitenbücher, Haupt, and Jäger. We further establish the decomposition formula $d={\sum }_{\mu \in {\mathcal{M}}_G^e(X)}{\iota }_{\mu }\exp (h_{\mu }^{\ast }(G)),$ where ${\iota }_{\mu }$ is the degree of the factor map from the measure-theoretic maximal compact factor associated with $\mu$ onto $X_{\mathrm{eq}}$, and $h_{\mu }^{\ast }(G)$ is the maximal measure sequence entropy of $\mu$. As further consequences of the decomposition formula, we show that for every finite $N$ with $2\le N\le d$, the system admits an essential IT $N$-tuple. Consequently, $h_{\mathrm{top}}^{\ast }(X,G)\ge \log d.$ This strengthens a lower bound of Liu, Wang, and Xu by also detecting compact multiplicities. As an application, we answer a question of Gómez, León-Torres, and Muñoz-López. If $G$ contains a finite-index normal subgroup isomorphic to ${\mathbb{Z}}^r$, then, for every $m\ge 2$, there exists a free minimal uniquely ergodic zero-entropy finite-alphabet $G$-subshift with maximal topological sequence entropy $\log m$, an essential IT $m$-tuple, and no essential IN $(m+1)$-tuple. For $G=\mathbb{Z}$, the alphabet may be chosen to have exactly $m$ symbols. Finally, we realize every finite multiplicity profile by a zero-entropy minimal almost one-to-one extension of an irrational circle rotation.

Added an application answering Question 5.1 of Gómez, León-Torres, and Muñoz-López concerning uniquely ergodic realizations of finite maximal sequence entropy. We welcome any comments, suggestions, or discussion regarding our manuscript

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