Variational principles of metric mean dimension for random dynamical systems

Abstract In this paper, we establish variational principles for the metric mean dimension of random dynamical systems with infinite topological entropy. This is based on four types of measure-theoretic ϵ -entropies: Kolmogorov-Sinai ϵ -entropy, Shapira’s ϵ -entropy, Katok’s ϵ -entropy and Brin–Katok local ϵ -entropy. The variational principle, as a fundamental theorem, links topological dynamics and ergodic theory.