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arXiv · 2607.01501

Uniformly Positive Mean Dimension

Tal Barak·Elon Lindenstrauss
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Abstract

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.

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Tal Barak, Elon Lindenstrauss. 2026-07-01. Uniformly Positive Mean Dimension. https://arxiv.org/abs/2607.01501

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