Topological Pressure For Nonautonomous Systems

In this thesis, we study the properties and applications of topology pressure fornonautonomous dynamical systems. Nonautonomous dynamical system is about a trajec-tory formed by a sequence of continuous self-maps of a compact metric space. Topologicalpressure for nonautonomous dynamical systems can be defined by spanning sets, separatedsets, or open covers respectively. We get some basic properties such as monotonicity, Lip-schitz continuity, convexity, etc and power law of topological pressure for nonautonomousdynamical systems. As applications, we get the commutativity of topological pressure forautonomous dynamical systems with respect to transformations. More precisely, for anycontinuous maps T and S from a compact metric space into itself, the maps T ? S andS ? T have the same topological pressure(with respect to the corresponding potentialfunctions).This thesis is organized as follows:In Chapter 1, the history and the development of topological pressure are presentedand the structure of the thesis is given at the end of this section.In Chapter 2, we introduce the definitions and properties of topological entropy andtopological pressure for autonomous dynamical systems, the definition and properties oftopological entropy for nonautonomous dynamical systems simply.Chapter 3 and Chapter 4, the main part of this thesis, include the result of myresearch.In Chapter 3, three kinds of equivalent definitions of topological pressure for nonau-tonomous dynamical systems are given, that is, by spanning sets, separated sets, or opencovers respectively. Because of the lack of subadditivity, the way of the definition of topo-logical pressure for nonautonomous dynamical systems is a little more complicated thanthat for autonomous dynamical systems.In Chapter 4, first of all, for a given nonautonomous dynamical system, we get howthe topological pressure changes when the potentials and the sequences of mappings vary.Then the monotonicity, Lipschitz continuity, convexity and other properties of topolog-ical pressure for nonautonomous dynamical systems are presented. Further, we obtainthe power rule of topological pressure for nonautonomous dynamical systems when thepotential function is constant function, the sequences of mappings are equicontinuous orthe sequences of mappings are periodic. Finally, as an application, the commutativity oftopological pressure for autonomous dynamical systems with respect to transformationsis obtained. This result is an improvement of the commutativity of topological entropyfor autonomous dynamical systems with respect to transformations.