Some Studies On The Complexity Of Topological Dynamical Systems

Topological dynamical system is an important part of nonlinear analysis,and it has many important applications in other scientific fields.There are a few methods,such as chaos,topological entropy and transitivity,to describe the complexity of a given system.This thesis mainly studies the complexity of topological dynamical systems and it is organized as follows:The first chapter is the introduction.It mainly summarizes the research status of topological dynamical systems.In the second chapter,we review some basic concepts,notations and several important conclusions which are needed in the thesis.In the third chapter,we improve and enrich the conclusions on the sequence asymptotically average tracing property by the recent results on tracing properties.In the fourth chapter,we introduce the concepts of weakly almost periodic point,measure center and minimal center of attraction of the actions on Amenable groups,and discuss the relationship between weakly almost periodic point,measure center and minimal center of attraction.Finally we explore the chaotic properties of such a system with a full measure center.In the fifth chapter,we raise the conception of family-R-T chaos and give some sufficient conditions for a system to be chaotic under some assumptions that the relating family fulfills different conditions.