Semigroup actions of expanding maps

TL;DR

This paper studies semigroup actions of Ruelle-expanding maps, analyzing their complexity through thermodynamic formalism, entropy, and zeta functions, and establishing foundational properties and measures for these dynamical systems.

Contribution

It introduces a framework connecting semigroup properties with thermodynamic formalism, including entropy, zeta functions, and stationary measures for Ruelle-expanding maps.

Findings

Topological entropy relates to periodic point growth.

Dynamical zeta function properties are characterized.

Existence of stationary probability measures is proven.

Abstract

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.