Specification and Thermodynamic Properties of Topological Time-Dependent Dynamical Systems

TL;DR

This paper explores the thermodynamic properties of time-dependent dynamical systems with the specification property, establishing conditions for positive entropy, chaos, and pressure computation, extending existing results to more general expanding maps.

Contribution

It introduces new conditions under which time-dependent systems have the specification property and positive entropy, extending prior results to Ruelle-expanding maps.

Findings

Systems with the specification property have positive topological entropy.

All points in these systems are entropy points, indicating topological chaos.

Conditions are provided for computing topological entropy and pressure as limits.

Abstract

This paper discusses the thermodynamic properties for certain time-dependent dynamical systems. In particular, we are interested in time-dependent dynamical systems with the specification property. We show that each time-dependent dynamical system given by a sequence of surjective continuous self maps of a compact metric space with the specification property has positive topological entropy and all points are entropy point. In particular, it is proved that these systems are topologically chaotic. We will treat the dynamics of uniformly Ruelle-expanding time-dependent dynamical systems on compact metric spaces and provide some sufficient conditions that these systems have the specification property. Consequently, we conclude that these systems have positive topological entropy. This extends a result of Kawan \cite{K2}, corresponding to the case when the expanding maps are smooth, to the more general case of expanding maps. Additionally, we study the topological pressure of time-dependent dynamical systems. We obtain conditions under which the topological entropy and topological pressure of any continuous potential can be computed as a limit at a definite size scale. Finally, we study the Lipschitz regularity of the topological pressure function for expansive (uniformly Ruelle-expanding) time-dependent dynamical systems on compact metric spaces.