Some Properties Of An Action Of An Amenable Group

In this paper,we consider the topological dynamical systems for amenable group actions.The text consists of the following six chapters.In Chapter 1,we introduce the historical background and recent development of the problems that will be studied in this thesis.In Chapter 2,we briefly review the definition and some properties of amenable group and subset density of amenable group.The main results of our study are presented in Chapters 3 to 5 of this thesis.In Chapter 3,we discuss the concept of Banach mean equicontinuity on compact metric spaces and some basic properties of Banach mean equicontinuity under the framework of amenable group actions,the unique ergodicity of the system for amenable group actions is also discussed.In particular,we give two equivalent conditions of the unique ergodicity for amenable group actions.Using unique ergodicity and the necessary conditions of Banach mean equicontinuity,we prove that the group action formed by the orbital closure corresponding to each point is uniquely ergodic in a Banach mean equicontinuity system for amenable group actions.Later,we study an important theorem,which play a key role in the proof of the main result in this chapter.In order to prove this theorem,we first introduce the concepts and the basic properties of generic point,the support of a measure and the support of a dynamical system for amenable group actions.Secondly,we analyze the relation between the support of a measure and the support of a dynamical system for amenable group actions.Finally,we prove two equivalent conditions of Banach proximal pairs.In the last part of this chapter,we obtain a main result by making use of unique ergodicity,Banach mean equicontinuity and the properties of Banach proximal relation under the framework of amenable group actions,we prove that if an action of an amenable group is Banach mean equicontinuous,then proximal relation,Banach proximal relation and regionally proximal relation are equivalent and it is a closed invariant equivalence relation.In Chapter 4,we introduce the Pesin-Pitskel topological pressure on an arbitrary subset of a compact metric space and give some important properties for amenable group actions.In particular,we prove that the definition of Pesin-Pitskel topological pressure will not be changed when “sup” at the part containing potential function is replaced by “inf”.This property is helpful to estimate the upper and lower bounds of Pesin-Pitskel topological pressure.Later,we introduce the Bowen pseudometric pressure on an arbitrary subset of a compact metric space and study some basic properties for amenable group actions.At the same time,the upper and lower bounds of the Bowen pseudometric pressure on subsets of an action of an amenable group can be estimated.Finally,we consider the local measure theoretic pressure and analyze some properties of local measure theoretic pressure for amenable group actions.In Chapter 5,we discuss the covering lemma for amenable group actions.The advantage of this lemma is that we can choose a countable sub-family consisting of pairwise disjoint Bowen balls satisfying some conditions from a family of Bowen balls.By applying this covering lemma,we study the relation between the Pesin-Pitskel topological pressure on subsets and the local measure theoretic pressure for amenable group actions,and calculate an example of the Pesin-Pitskel topological pressure on some subsets of Bernoulli shifts for amenable group actions.In Chapter 6,we summarize the main results of this thesis,and on this basis,we give a brief description of some issues to be studied in the future.