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

Abstract

We introduce the concept of upper metric mean dimension of a one-parameter family of scaled pressure functions, which extends the corresponding notion for a single potential and satisfies a variational principle. This approach, supported by Convex Analysis, conveys a definition of measure-theoretic upper metric mean dimension, which is concave and upper semi-continuous, and therewith equilibrium states. In the context of dynamical systems, we establish a variational principle for the metric mean dimension with potential in terms of Katok entropy. As an application, we provide a simple formula for the upper metric mean dimension with potential for the shift on the space ([0,1]D N, for every D is an element of N, which links mean dimension theory with ergodic optimization.