Some Researches On Sensitivity And Transitivity Of Topological Flows

A topological(semi-)flow is a dynamical system which means that a(semi-)group continuously actions on a compact metric space.This thesis mainly studies the relationships between the sensitive sets and regionally proximal sets of topological flows,as well as the relationships between several kinds of transitivity of the dynamical system which is generated by a free semi-group continuously acting on a compact metric space and its shew product system.The full thesis is divided into four chapters.Chapter 1 is an introduction in which we recall the backgrounds,motivations and current developments of topological dynamical systems related to this thesis,and briefly review some related concepts of discrete topological semi-dynamical systems.Chapter 2 is to study mainly the relationships of sensitivity and S-set,S-set and Q-set for transitive topological flows,as well as by the cardinalities of S-sets,the sensitivity of transitive topological flows is divided into several kinds and the characteristics of these sensitivities and their relations are investigated.Chapter 3 is to consider some dynamics of free semi-group actions.At first,the concepts of multi-transitivity,?-transitivity and ?-mixing for a free semi-group action are introduced and the relationships among them are explored.Moreover,the relationships of such properties between the free semi-group action and its shew product system are also revealed.Secondly,the notions of topological entropy set and entropy string and entropy pair are extended to the case of free semi-group actions and the conclusions that the system with topological entropy string or entropy pair or entropy set must have positive topological entropy are obtained.Finally,the relationships among these three concepts and their topological invariance under factor maps are also discussed.Chapter 4 is the summary and prospect in which we put forward some problems to be discussed in the future.