  We consider semigroups of Ruelle-expanding maps, parameterized by random
walks on the free semigroup, with the aim of examining their complexity and
exploring the relation between intrinsic properties of the semigroup action and
the thermodynamic formalism of the associated skew-product. In particular, we
clarify the connection between the topological entropy of the semigroup action
and the growth rate of the periodic points, establish the main properties of
the dynamical zeta function of the semigroup action and prove the existence of
stationary probability measures.
