Abstract
We consider continuous free semigroup actions generated by a family
$(g_y)_{y \,\in \, Y}$
of continuous endomorphisms of a compact metric space
$(X,d)$
, subject to a random walk
$\mathbb P_\nu =\nu ^{\mathbb N}$
defined on a shift space
$Y^{\mathbb N}$
, 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
$Y^{\mathbb {N}}$
, 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.
Topological Entropy of Free Semigroup Actions Generated by Proper Maps for Noncompact Subsets
