Some Researches On A Flow And Its Induced Flows And Local Entropy

By a flow we mean that a topological group jointly continuously actions on a compact metric space.Some properties of a semi-discrete topological dynamical system can be generalized to flows.This thesis mainly studies the relationships among some sensitivities and some transitivities of the original flow and its induced flows,explores the connections of levels of the orbit's topological structure and their topological entropy of an Amenable group action.The whole thesis is divided into four chapters as follows:Chapter 1 is an introduction in which the background of topological dynamical systems and the current research situation are presented.Moreover,the related concepts of topological dynamical systems are summarized.In chapter 2,let(T,X)be a flow,that is X is a compact metric space and T is a group action on X and let K(X),M(X)and F(X)be the sets consisting of all compact subsets of X,all probability Borel measures on X and all upper semi-continuous fuzzy sets on X,respectively.And K(X),M(X),IF(X)are metric spaces under Hausdorff metric,Prohorov metric and level-wise metric respectively.So the group can be naturally regarded as an action on K(X),M(X),F(X)and generates new flows(T,K(X)),(T,M(X))and(T,F(X)).The aim of this chapter is to investigate the relations of some sensitivities and some transitivities of the original flow(T,X)and its induced flows(T,K(X)),(T,M(X))and(T,IF(X)).Chapter 3 mainly studies the dynamic properties of discrete infinitely countable Amenable group action.Concretely,we define some new notions of recurrent points by using the Folner sequence and study their relations.The topological entropy of all kinds of point sets is also discussed.Chapter 4 is a summary and prospect in which some unresolved problem are posed.