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

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