AI 中文总结
本文针对有限开覆盖建立了平均维数与拓扑熵及基数的定量比较，证明零熵覆盖具有零平均维数，并推广到 amenable 群及 sofic 情形，进而得出平均维数对为熵对且 UPMD 蕴含 UPE，回答了相关开放问题。
AI 中文摘要
我们为单个有限开覆盖建立了Lindenstrauss和Weiss的全局熵-平均维数比较的对应结果。更精确地说，有限开覆盖的平均维数由其拓扑熵和基数定量地界定。特别地，具有零熵的有限开覆盖具有零平均维数。同样的比较也适用于可数 amenable 群的作用以及 sofic 情形。由此得出，每个平均维数对都是熵对，并且特别地，一致正平均维数（UPMD）蕴含一致正熵（UPE）。覆盖和对的蕴含分别回答了García-Ramos和Gutman的问题2.27和4.9。证明将一个多面体与一个最小子覆盖相关联，并将Lindenstrauss-Weiss压缩论证与Maurey的经验方法相结合来估计其坐标投影的体积。
英文摘要
We establish a counterpart, for individual finite open covers, of the global entropy-mean-dimension comparison of Lindenstrauss and Weiss. More precisely, the mean dimension of a finite open cover is quantitatively bounded in terms of its topological entropy and cardinality. In particular, a finite open cover with zero entropy has zero mean dimension. The same comparison holds for actions of countable amenable groups and in the sofic setting. It follows that every mean dimension pair is an entropy pair and, in particular, that uniformly positive mean dimension (UPMD) implies uniformly positive entropy (UPE). The cover and pair implications answer Questions 2.27 and 4.9 of García-Ramos and Gutman, respectively. The proof associates a polytope to a minimal subcover and combines the Lindenstrauss-Weiss compression argument with Maurey's empirical method to estimate the volumes of its coordinate projections.
Comments17 pages
