Some Researches On Several Dynamic Properties Of Topological Dynamical Systems

The complexity of dynamic properties such as strongly mixing property,chaoticity, topological entropy, specification properties are hot topics in the study of dynamical systems, which describe the complexity of dynamical systems in different ways. This thesis mainly focuses on the studies of several dynamic properties of topological dynamical systems. The whole thesis is divided into four chapters.Chapter 1 is an introduction which summarizes the historical background of topological dynamical systems and current status of the related researches. Moreover,some basic concepts of topological dynamical system are recalled.Chapter 2 aims to study the complexity of symbol dynamic systems containing proper quasi-weakly almost periodic points. For instance, are they strongly mixing,chaotic or how about topological entropy? etc. After generalization, some sufficient conditions for a topological dynamical system to have positive topological entropy to be strongly mixing and strongly distribution chaos are proved, respectively.The purpose of Chapter 3 is to discuss several specification properties and distinguish the inclusion relationship among them. Furthermore, two weak specification properties are introduced, which are called quasi-weak specification property and semi-weak specification property in the thesis. By constructing method,we show that quasi-weak specification property and semi-weak specification property are strictly weaker than specification property. What is more, we prove the equivalence of quasi-weak specification property, semi-weak specification property and strongly mixing.In Chapter 4, we consider an open problem related to quasi-weakly almost periodic points in flows which was firstly posed in discrete dynamical systems. We construct examples to show that the open problem also has a positive answer in the case of flows.