Some Studies On Chaotic Dynamics Of Amenable Group Actions

In this thesis,we mainly generalized some concepts and absorbing results of sensitive dependence and equicontinuity of discrete dynamic systems to countable discrete amenable group actions(amenable group actions for short)and improved some main conclusions in relevant paper.This thesis is divided into six chapters,the specific contents are listed as follows:In Chapter 1,we mainly reviewed the historical backgrounds and research status of topological dynamic systems,as well as some necessary preparatory knowledge.In Chapter 2,we firstly studied the relationship between different equicontinuity for amenable group actions,gave the corresponding dichotomy theorem and some sufficient conditions for an amenable group action to be the density-n-sensitive with respect to a F(?)lner sequence,and we proved that each topological sequence n-entropy tuple is a density-sensitive n-tuple with respect to every tempered F(?)lner sequence for an abelian group action which admits an ergodic measure with full support.In Chapter 3,we proved that every measure-theoretical-sequence n-entropy tuple under some given conditions is a measure-theoretical-density-n-sensitive tuple with respect to any given tempered F(?)lner sequence.In Chapter 4,we firstly introduced the concept of measure-theoretical-mean n-sensitive tuple for an amenable group action,and obtained that every measuretheoretic n-entropy tuple is a measure-theoretical-mean n-sensitive tuple under some given conditions.Then,we introduced the concept of M-ergodicity of order n for an amenable group action and obtained their equivalence relations with the measure-theoretical mean n-sensitive tuple.In Chapter 5,we mainly studied the properties of mean n-sensitive tuple and weakly n-sensitive in the mean tuple for amenable group actions and obtained the relationship between the topological n-entropy tuple and the mean n-sensitive tuple under some given conditions.Then,we introduced the concept of M-transitivity of order n for an amenable group action and obtained their equivalence relations with the mean n-sensitive tuple under some given conditions.We proposed the concept of the maximal mean equicontinuous factor for an amenable group action and presented some relevant conclusions.Chapter 6 is a conclusion and prospect of this thesis.First of all,in this chapter,we summarized the main conclusions of this thesis and put forward several questions for the future researches.