Some Studies On The Complexities Of Topological Dynamical Systems

In the present thesis, some complexities including topological transitivity,Devaney's chaos and sensitivity, of topological dynamical systems are studied. The thesis is organized as follows:In chapter 1, we briefly recall the origin and development of topological dynamical systems, and state the source and background of the problems discussed in this thesis.In chapter 2, we review some basic concepts and notions on topological dynamical systems, as well as some known properties related to our studies on topological dynamical systems.In chapter 3, we introduce the conception of the sequence asymptotic average shadowing property, which is a generalization of the asymptotic average shadowing property. In this chapter, the new property is used to investigate topological transitivity.In chapter 4, the relations among transitivity, indecomposability and Z- transitivity are discussed. Besides, a new transitive level as weak transitivity is introduced and some equivalent conditions of Devaney's chaos are given by weak transitivity. Moreover, it is proved that both (?)-shadowing property and (?)-shadowing property imply weak transitivity.In chapter 5, we firstly introduce several stronger forms of sensitivities in the dynamical systems of semigroup actions, such as thick sensitivity and thickly syndetical sensitivity, and obtain some sufficient conditions for a dynamical system to have such sensitivities. We prove that a weakly mixing system of semigroup actions is thickly sensitive and a minimal weakly mixing system as well as a nonminimal M-system of semigroup actions is thickly syndetically sensitive.