A variational principle for the metric mean dimension of free semigroup actions

Abstract

We consider continuous free semigroup actions generated by a family (g(y))(y) (epsilon) (Y) of continuous endomorphisms of a compact metric space (X, d), subject to a random walk P-nu = nu(N) defined on a shift space Y-N, where (Y, d(Y)) is a compact metric space with finite upper box dimension and nu is a Borel probability measure on Y. With the aim of elucidating the impact of the random walk on the metric mean dimension, we prove a variational principle which relates the metric mean dimension of the semigroup action with the corresponding notions for the associated skew product and the shift map sigma on Y-N, and compare them with the upper box dimension of Y. In particular, we obtain exact formulas whenever nu is homogeneous and has full support. We also discuss several examples to enlighten the roles of the homogeneity, of the support and of the upper box dimension of the measure nu, and to test the scope of our results.