Specification and Thermodynamic Properties of Topological Time-Dependent Dynamical Systems

  • Javad Nazarian Sarkooh
  • F. H. GhaneEmail author


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 (Nonlinearity 28:669–695, 2015), 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 and hence for uniformly Ruelle-expanding time-dependent dynamical systems on compact metric spaces.


Non-autonomous dynamical system Topological entropy Specification property Entropy point Topological pressure Ruelle-expanding map 

Mathematics Subject Classification

37B55 37C60 37C40 37B40 37A25 



The authors would like to thank the respectful referee for his/her comments on the manuscript.


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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of MathematicsFerdowsi University of MashhadMashhadIran

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