Double variational principle for mean dimension

Abstract

We develop a variational principle between mean dimension theory and rate distortion theory. We consider a minimax problem about the rate distortion dimension with respect to two variables (metrics and measures). We prove that the minimax value is equal to the mean dimension for a dynamical system with the marker property. The proof exhibits a new combination of ergodic theory, rate distortion theory and geometric measure theory. Along the way of the proof, we also show that if a dynamical system has the marker property then it has a metric for which the upper metric mean dimension is equal to the mean dimension.

Symbolic dynamics in mean dimension theory

“…Here is one remark about the notation. In the paper [LT19], the lower mean Hausdorff dimension played no role. So the upper mean Hausdorff dimension was simply denoted by mdim H (X , T, d) in [LT19].…”

“…Mean dimension theory provides a meaningful framework for extending (1.1) to higher rank actions. This is the theory first introduced by Gromov [Gro99] and further developed by Lindenstrauss and Weiss [LW00], Lindenstrauss [Lin99] and more recently Lindenstrauss and the second named author [LT19]. We review the basic ingredients here.…”

“…This is a dynamical analogue of Minkowski dimension. Lindenstrauss and the second named author [LT19] defined mean Hausdorff dimension mdim H (X , T, d). This is a dynamical analogue of Hausdorff dimension.…”

Variational relations for metric mean dimension and rate distortion dimension

“…By R µ ( ), R µ,L ∞ ( ) we denote the rate distortion function and the L ∞ -rate distortion function, respectively (see Section 2 for the definitions). For µ ∈ M (X, T ), the upper rate distortion dimension studied by Lindenstrauss and Tsukamoto [9] is defined as…”

“…Besides, Velozo-Velozo [15] proved a corresponding variational principle by using the Katok mesure-theoretic entropy (defined by spanning sets) instead of the rate distortion function. In 2019, Lindenstrauss and Tsukamoto [9] constructed µ ∈ M (X, T ) capturing dynamical complexity of (X, T ) over all resolution > 0 and established the following double variational principle between mean dimension and upper rate distortion dimension. Theorem 1.3.…”