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We consider continuous free semigroup actions generated by a family (gy)y∈Y of continuous endomorphisms of a compact metric space (X, d), subject to a random walk Pν = νN defined on a shift space Y N, where (Y, dY ) is a compact metric space with finite upper box dimension and ν 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 σ on Y N, and compare them with the upper box dimension of Y . In particular, we obtain exact formulas whenever ν 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 ν, and to test the scope of our results.

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