Research Of The Properties Of F-mixing

Recurrence in Ferstenberg family is one of the important parts in topological dynamics. In 1981,in the artical [1],by the subsets of Z or Z+,Hillel Furstenberg first described how frequently the orbits of a point x returned into its neighbourhood ,then he got the notion of Furstenberg family.In the artical [2],E.Akin researched the theory in detail and explicitly described this useful piece of mathematical folklore . This paper mainly researchs on the F-mixing with respect to a family F.It expands the properties of topological double ergodic to F-mixing .At the beginning of this paper ,we are devoted to studying the equivalent conditions of the F-mixing and of chaotic subsets. Later we research on the property of KtB mixing . There are four chapters:In chapter one ,it is preliminary knowledge.We mainly introduce the conceptions that will be used in this paper and main conclusions we get.In chapter two ,we wide the corresponding results of Yang[3],ie: Lemma 1,Lemma 2,Lemma 3 to F-mixing and get theorem 2.1,theorem 2.2,theorem 2.3.Then we discuss factor map and inverse limit space and get theorem 2.4,theorem 2.5 and theorem 2.6.In chapter three,on a continuous map of the compact metric space into itself, we discuss the equivalent conditions between F-mixing and chaotic subset by Liao and Xiong.We also research the relation on F-mixing and distribute chaotic sets,then get corresponding conclusions.In chapter four,for KTB mixing,we get its properties and judging.