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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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Cite

BibTeXRIS
BibTeX
@misc{barak2026uniformly,
  title = {{Uniformly Positive Mean Dimension}},
  author = {Tal Barak and Elon Lindenstrauss},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2607.01501},
  eprint = {2607.01501},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2607.01501},
}
RIS
TY  - UNPB
TI  - Uniformly Positive Mean Dimension
AU  - Tal Barak
AU  - Elon Lindenstrauss
PY  - 2026
DA  - 2026/09/30
PB  - arXiv
DO  - 10.48550/arxiv.2607.01501
UR  - https://arxiv.org/abs/2607.01501
AB  - 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.
ER  -