Some Studies On The Shadowing Property Of Topological Dynamical Systems

The shadowing property is an important concept in topological dynamical systems and plays a key role in the developments of topological dynamical systems.In this thesis,the notion of shadowable points of set-valued dynamical systems is firstly introduced and some basic properties of shadowable points are presented,then it is proved that the totally transitive property is equivalent to the mixing property for set-valued dynamical systems with the shadowing property.Finally,for discrete topological dynamical systems,the concept of sequence almost average shadowing property is introduced and some dynamical properties of such systems with the sequence almost average shadowing property are obtained.The detailed arrangement is as follows:The first chapter is an introduction,which briefly describes the current research status of the shadowing property of dynamical systems and set-valued dynamical systems,and simply explains the research background and source of questions considered in this thesis.In the second chapter,we firstly introduce the conception of shadowable points in set-valued dynamical systems and prove some related properties of shadowable points.Then we explore some equivalent conditions for a set-valued dynamical system with the shadowing property to be mixing.Finally we introduce the average shadowing property in set-valued dynamical systems and prove every set-valued dynamical system has the average shadowing property if it has the specification and the shadowing property.In the third chapter,we introduce the concept of the sequence almost average shadowing property in discrete topological dynamical systems and show some dynamical properties of such systems.The fourth chapter is the conclusion and outlook in which we summarize the conclusions of this thesis and propose some problems to be solved.