Some Studies On The Eventual Shadowing Property Of Topological Dynamical Systems

Eventual shadowing property is a new concept that was recently introduced in the studies of topological dynamical systems,which plays an important role in the studies of the complexity of dynamical systems.In this thesis,some studies are carried out on the complexity of set-valued dynamical systems and topological dynamical systems with the eventual shadowing property.Concretely,it is proved that a set-valued dynamical system with chain transitivity either has no eventually shadowable points or all points in the system are eventually shadowable points,and if a topological dynamical system with the eventual shadowing property contains an eventually shadowable point which is recurrent but not almost periodic,then the system has positive entropy,etc.The specific arrangements are as follows:Chapter 1 is an introduction.Firstly,the current situation of topological dynamical systems is briefly described.Secondly,several types of shadowing properties are introduced.Then,the development of eventual shadowing property of set-valued dynamical systems is summarized.Finally,the research background and source of the problems studied in this paper are presented.In the second chapter,we firstly introduce the definitions of eventually shadowable point and eventual shadowing property for set-valued dynamical systems and prove some recurrent properties and chaotic properties for the systems with eventual shadowing property.Then,we obtain some equivalent conditions for a set-valued dynamical system with the eventual shadowing property to be topologically mixing.In the third chapter,we mainly study the complexity of topological dynamical systems with the eventual shadowing property and obtain some sufficient conditions for such systems to have positive entropy and some equivalent conditions for such system to be topologically mixing.Chapter 4 is the review and prospect of this thesis in which we summarize the main conclusions of the whole thesis,and put forward some unsolved problems and some views on the future development trend of the research direction of this thesis.