An Entropy Bound for Mean Dimension of Open Covers
Abstract
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.
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BibTeX
@misc{barak2026entropy,
title = {{An Entropy Bound for Mean Dimension of Open Covers}},
author = {Tal Barak},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.08625},
eprint = {2610.08625},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.08625},
}
RIS
TY - UNPB
TI - An Entropy Bound for Mean Dimension of Open Covers
AU - Tal Barak
PY - 2026
DA - 2026/10/06
PB - arXiv
DO - 10.48550/arxiv.2610.08625
UR - https://arxiv.org/abs/2610.08625
AB - 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.
ER -