Specification properties and thermodynamical properties of semigroup actions

TL;DR

This paper investigates the thermodynamical properties of finitely generated semigroup actions, introducing new notions of entropy and specification properties that reveal their dynamic complexity and relationships with periodic points.

Contribution

It introduces strong and orbital specification properties for semigroup actions and studies their implications for entropy, pressure functions, and periodic points.

Findings

Semigroup actions can have positive topological entropy under certain specification properties.

Pressure functions exhibit convergence and Lipschitz regularity.

Relations between entropy and exponential growth rate of periodic points are established.

Abstract

In the present paper we study the thermodynamical properties of finitely generated continuous subgroup actions. We address a notion of topological entropy and pressure functions that does not depend on the growth rate of the semigroup and introduce strong and orbital specification properties, under which, the semigroup actions have positive topological entropy and all points are entropy points. Moreover, we study the convergence and Lipschitz regularity of the pressure function and obtain relations between topological entropy and exponential growth rate of periodic points in the context of semigroups of expanding maps. The specification properties for semigroup actions and the corresponding one for its generators and the action of push-forward maps is also discussed.