Around the variational principle for metric mean dimension

10/27/2020

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by   Yonatan Gutman, et al.

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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 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 and we investigate them also for individual measures. Finally we develop a lower bound for metric mean dimension in terms of Brin-Katok local entropy.