Some Studies On Sensitivity And Ergodicity Of Amenable Group Actions

This thesis mainly extends some ergodic theories and transitive properties of discrete dynamical systems to countable discrete amenable group actions（amenable group actions for short）,and gets some sufficient conditions（necessary and sufficient conditions）for an amenable group action to possess a certain sensitivity and ergodicity.The whole thesis is divided into four chapters,the specific contents are as follows:Chapter 1 is an introduction which mainly introduces the relevant backgrounds and current research situations of discrete dynamical systems as well as some preliminary knowledge for the studies of this thesis.In Chapter 2,we prove firstly several instrumental ergodic conclusions of amenable group actions,then for a F（?）lner sequence{F _n}^?_n＝1of a given amenable group,we show that the nonS-genericminimal center of attraction relative to{F _n}^?_n＝1of a point is sensitive.In addition,we obtain that the nonS-generic minimal center of attraction relative to{F _n}^?_n＝1of a point with dense quasi-weakly almost periodic point is syndeticly sensitive.In Chapter 3,we firstly study the internal structure of the recurrent level of proper quasi-weakly almost periodic points relative to a given F（?）lner sequence{F _n}^?_n＝1of amenable group actions with the strong specification property and weak specification property,respectively,then for the systems with the strong specification property,we obtain that the set of proper quasi-weakly almost periodic points relative to a given F（?）lner sequence{F _n}^?_n＝1satisfying“there exists an invariant measure generated by the orbit of each point along{F _n}^?_n＝1such that its support is just the minimal center of attraction relative to{F _n}^?_n＝1”is a residual set.In Chapter 4,we mainly study that ergodic mixing,topologically double ergodic property,weak mixing,extreme scattering and strong scattering are equivalent for the amenable group actions with a full measure center.Chapter 5 is a review and prospect of this thesis.Firstly,we recall the main conclusions of this thesis,then we put forward some problems for the further research.