Around the variational principle for metric mean dimension
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.
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