2026/07/29 by Chunlin Liu
Mathematics · #math.DS
We welcome any comments, suggestions, or discussion regarding our manuscript
arxiv created 2026/07/29 · arxiv updated 2026/07/31
Let G be a countably infinite discrete amenable group acting minimally on a compact metric space X, and let π:X→ Xeq be the maximal equicontinuous factor map. Let d∈\mathbb N∪\∞\ be the conditional topomorphic degree; when d<∞, it is the least integer such that π is an at most d-to-one topomorphic extension. We prove that, for every r≥2, the following are equivalent: the system is Weyl mean r-equicontinuous; it is mean r-equicontinuous along some Følner sequence; and d≤ r-1. For minimal \mathbb Z-systems, this resolves a conjecture of Breitenbücher, Haupt, and Jäger. We establish the formula d=∑μ∈\mathcal MGe(X)ιμexp(hμ^*(G)), where ιμ is the degree from the measure-theoretic maximal compact factor associated with μ onto Xeq, and hμ^*(G) is maximal measure-theoretic sequence entropy. Consequently, every finite N with 2≤ N≤ d yields an essential IT N-tuple, and htop^*(X,G)≥log d. This strengthens the known sequence-entropy lower bound by also detecting the compact multiplicities ιμ. Finally, every finite multiset of positive-integer pairs \(ι1,b1),…,(ι_ℓ,b_ℓ)\ is realized by a zero-entropy minimal almost one-to-one extension (X,T) of an irrational circle rotation with exactly ℓ ergodic invariant measures μ1,…,μ_ℓ satisfying ιμi=ιi and exp(hμi^*(\mathbb Z))=bi for 1≤ i≤ℓ. The resulting conditional topomorphic degree is ∑i=1ℓιi bi.