AI 中文总结
研究固定开覆盖下的一致正熵与一致正平均维数的关系，通过构造轮毂-辐条系统，证明完全支撑不变测度或一致正熵可推出一致正平均维数，并给出反例。
AI 中文摘要
我们研究了在固定开覆盖层面上的一致正熵与一致正平均维数之间的关系。对于一个符号系统X，我们关联一个轮毂-辐条系统Spoke(X)，该系统的构造方法是将每个符号替换为一个附着在公共轮毂上的一维辐条。我们证明，如果X存在一个全支撑的移位不变测度，那么Spoke(X)具有完全正平均维数。我们还证明，如果X具有一致正熵，那么Spoke(X)具有一致正平均维数。最后，利用环面上无理旋转的符号编码，我们构造了具有完全正平均维数但不具有一致正平均维数或一致正熵的轮毂-辐条系统。这些例子是非退化的：相关的覆盖具有零平均维数和零熵，但在动力学迭代下细化时，相应的覆盖数是无界的。
英文摘要
We study the relation between uniformly positive entropy and uniformly positive mean dimension at the level of fixed open covers. To a symbolic system X, we associate a hub-and-spoke system Spoke(X), obtained by replacing each symbol by a one-dimensional spoke attached to a common hub. We prove that if X admits a shift-invariant measure of full support, then Spoke(X) has completely positive mean dimension. We also prove that if X has uniformly positive entropy, then Spoke(X) has uniformly positive mean dimension. Finally, using symbolic codings of irrational rotations on tori, we construct hub-and-spoke systems with completely positive mean dimension but without uniformly positive mean dimension or uniformly positive entropy. The examples are nondegenerate: the relevant covers have zero mean dimension and zero entropy, but when refined by iterating under the dynamics the corresponding covering numbers are unbounded.
Comments32 pages. Minor revisions; added a remark and example concerning the hub-and-spoke construction. Results unchanged
