Around the variational principle for metric mean dimension

Abstract

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.

Some Notes on Variational Principle for Metric Mean Dimension

“…The following variational principle for the metric mean dimension was obtained by Gutman and Śpiewak [7].…”

“…Proof. Let U be a finite open cover of X with diameter diam(U) ≤ 4ε * and Lebesgue number Leb(U) ≥ ε * , the existence of such U is guaranteed by [7,Lemma 3.4]. By Ergodic Decomposition Theorem, there exists a measure μ on M f (X) satisfying μ(M e f (X)) = 1 such that µ = M e f (X) τ dμ(τ ).…”

Bowen’s equations for upper metric mean dimension with potential

“…Very recently, Lindenstrauss and Tsukamoto's pioneering work [LT18] showed a first important relationship between mean dimension theory and ergodic theory, which is an analogue of classical variational principle for topological entropy. More discussions associated with this result can be found in [GS21,CDZ22]. From that time on, Lindenstrauss and Tsukamoto's work inspired more and more researchers to inject ergodic theoretic ideas into mean dimension theory by constructing some new variational principles, and we refer to [VV17, LT19, Tsu20, GS21, Shi21, CLS21, W21] for more details.…”

A convex analysis approach to the metric mean dimension: Limits of scaled pressures and variational principles

“…Actually, Y. Gutman and A. Śpiewak showed in [11] that it suffices to take the previous supremum over E T (X), and obtained a new variational principle linking the upper metric mean dimension to the metric entropy h µ , namely mdim M (X, d, T ) = lim sup…”