AI 中文总结
本文针对极小顺从群作用，建立了条件拓扑度与均值等度连续性的等价关系，解决了相关猜想，给出了条件拓扑度的公式，还实现了特定多重集对应的扩张并强化了序列熵下界
AI 中文摘要
设G是可数无限离散顺从群，极小地作用在紧度量空间X上，且令π:X→X_eq为极大等连续因子映射。设d∈N∪{∞}为条件拓扑度；当d<∞时，它是使得π为至多d对一的拓扑扩张的最小整数。我们证明，对每个r≥2，以下条件等价：该系统是Weyl均值r-等连续的；它沿某个Følner序列是均值r-等连续的；且d≤r-1。对于极小ℤ-系统，这解决了Breitenbücher、Haupt和Jäger的一个猜想。我们建立公式d=∑_{μ∈𝒫_G^e(X)}ι_μexp(h_μ^*(G))，其中ι_μ是与μ相关联的测度论极大紧因子到X_eq的次数，h_μ^*(G)是极大测度论序列熵。因此，每个满足2≤N≤d的有限N都产生一个本质IT N-元组，且h_top^*(X,G)≥log d。这通过同时检测紧重数ι_μ，强化了已知的序列熵下界。最后，每个由正整数对构成的有限多重集{ (ι₁,b₁),…,(ι_ℓ,b_ℓ) }都可由无理圆周旋转的零熵极小几乎一对一扩张(X,T)实现，该扩张恰有ℓ个遍历不变测度μ₁,…,μ_ℓ，使得对1≤i≤ℓ，满足ι_{μ_i}=ι_i且exp(h_{μ_i}^*(ℤ))=b_i。所得的条件拓扑度为∑_{i=1}^{ℓ}ι_i b_i。
英文摘要
Let $G$ be a countably infinite discrete amenable group acting minimally on a compact metric space $X$, and let $π:X\to X_{\mathrm{eq}}$ 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 $π$ is an at most $d$-to-one topomorphic extension. We prove that, for every $r\ge2$, 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 $d=\sum_{μ\in\mathcal M_G^e(X)}ι_μ\exp\bigl(h_μ^*(G)\bigr),$ where $ι_μ$ is the degree of the factor map from the measure-theoretic maximal compact factor associated with $μ$ onto $X_{\mathrm{eq}}$, and $h_μ^*(G)$ is the maximal measure sequence entropy of $μ$. 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, $h_{\mathrm{top}}^*(X,G)\ge \log d.$ 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 $m\ge2$, 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.
CommentsAdded 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
