Ergodic Theory For Amenable Group Actions

This paper mainly studies the knowledge system and structure exhibited by the ergodic theory under the action of the amenable group and the integer group.It is mainly manifested by the form of ergodic theory around some basics,from the role of integer group to the form of amenable group.The first part,we introduce some basic concepts and definitions of topological dynamic systems and ergodic properties under the action of integer groups.Some related theorems and detailed proofs about ergodic theory and Poincaré's reply theorem are given.The second part is corresponding to the basic concepts,definitions and theorems of the ergodic theory under the action of integer groups in the first part,which are extended to the dynamic system under the action of the amenable group.Specifically,it includes the Poincaré reply theorem,the definition of ergodicity,the equivalent proposition and proof of ergodicity,the mean ergodic theorem and the proof and pointwise ergodic theorem and proof.The third part mainly studies the weak mixing of the dynamical system,and introduces the proposition equivalent to the weak mixing,and gives a detailed proof.Finally,it introduces a definition of the weak mixing of the dynamical system under the action of the amenable group.