Around the variational principle for metric mean dimension

Yonatan Gutman, Adam Spiewak

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We study variational principles for metric mean dimension. First we prove that in the variational principle of Lindenstrauss and Tsukamoto it suffices to take the supremum over ergodic measures. Second we derive a variational principle for metric mean dimension involving growth rates of measure-theoretic entropy of partitions decreasing in diameter which holds in full generality and in particular does not necessitate the assumption of tame growth of covering numbers. The expressions involved are a dynamical version of Rényi information dimension. Third we derive a new expression for the Geiger-Koch information dimension rate for ergodic shift-invariant measures. Finally we develop a lower bound for metric mean dimension in terms of Brin-Katok local entropy.

Original languageEnglish
Pages (from-to)345-360
Number of pages16
JournalStudia Mathematica
Issue number3
StatePublished - 2021
Externally publishedYes

Bibliographical note

Publisher Copyright:
© 2021 Institute of Mathematics. Polish Academy of Sciences. All rights reserved.


  • Information dimension
  • Metric mean dimension
  • Variational principles


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