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

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. Let $d\in\mathbb N\cup\{\infty\}$ be the conditional topomorphic degree; when $d<\infty$, it is the least integer such that $\pi$ is an at most $d$-to-one topomorphic extension. We prove that, for every $r\ge2$, the following are equivalent: the system is Weyl mean $r$-equicontinuous; it is mean $r$-equicontinuous along some F{\o}lner sequence; and $d\le r-1$. For minimal $\mathbb Z$-systems, this resolves a conjecture of Breitenb\"ucher, Haupt, and J\"ager. We establish the formula $d=\sum_{\mu\in\mathcal M_G^e(X)}\iota_\mu\exp(h_\mu^*(G))$, where $\iota_\mu$ is the degree from the measure-theoretic maximal compact factor associated with $\mu$ onto $X_{\mathrm{eq}}$, and $h_\mu^*(G)$ is maximal measure-theoretic sequence entropy. Consequently, every finite $N$ with $2\le N\le d$ yields an essential IT $N$-tuple, and $h_{\mathrm{top}}^*(X,G)\ge\log d$. This strengthens the known sequence-entropy lower bound by also detecting the compact multiplicities $\iota_\mu$. Finally, every finite multiset of positive-integer pairs $\{(\iota_1,b_1),\ldots,(\iota_\ell,b_\ell)\}$ is realized by a zero-entropy minimal almost one-to-one extension $(X,T)$ of an irrational circle rotation with exactly $\ell$ ergodic invariant measures $\mu_1,\ldots,\mu_\ell$ satisfying $\iota_{\mu_i}=\iota_i$ and $\exp(h_{\mu_i}^*(\mathbb Z))=b_i$ for $1\le i\le\ell$. The resulting conditional topomorphic degree is $\sum_{i=1}^{\ell}\iota_i b_i$.