Some Studies On Complexity Of Group Action Systems

The concept of equicontinuity is frequently used in the description of the complex-ity of topological dynamical systems.Sensitivity as the opposite-side of equicontinuity is also an important concept,which is usually used to study some chaotic properties of topological dynamical systems.This thesis includes two parts:the first one is to study the equicontinuity of group actions;the second is to introduce two new recurrent levels(weakly almost periodic point and quasi-weakly almost periodic point)of Z~d+-actions and to investigate the chaotic property of the minimal center of attraction of a point of Z~d+-action.The thesis is divided into five chapters,the concrete contents are stated as follows:The first chapter is an introduction and preparatory knowledges,which briefly summarizes the research status of topological dynamical systems;and introduces the current situation and sources of the research problems of this thesis;and some prepara-tory knowledge needed in this thesis are introduced.In the second chapter,the concepts of Hausdorff equicontinuity and Hausdorff sensitivity are firstly introduced and some conclusions of them are proved.Secondly,the concepts of topological equicontinuity and even continuity are introduced,and some properties of them are obtained.Moreover,it is proved that a minimal system is either topologically equicontinuous or Hausdorff sensitive under the assumption that the base space is a T3 space.In the third chapter,under the condition that the base space is a compact Hausdorff space,the concepts of rigidity and measure-theoretic equicontinuity with respect to a function is introduced,and some equivalent descriptions for a function to be rigid are given.In particular,in this chapter,some equivalent propositions for the system to be uniformly rigid are presented.In the fourth chapter,the concepts of(quasi-)weakly almost periodic point and minimal center of attraction for Z~d+-actions are introduced and the connections of levels of the topological structure of the orbits of(quasi-)weakly almost periodic points are explored.Moreover,the relations between(quasi-)weakly almost periodic point and minimal center of attraction are discussed.Especially,the chaotic dynamics near or inside the minimal center of attraction of a point are investigated.The fifth chapter mainly summarizes the conclusions of this paper and presents some issues for the future researches.