Family Of F-traversal, Swept Through The F-and F-mixed Nature Of The Study

Recurrence in Furstenberg 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 orbit of a point x returned into its neighbourhood,then he got the notion of family F.In the article [2],E.Akin researched some properties of family F profoundly.This paper mainly researches on the F-ergodic, F-mixing and F-sweep out with respect to a family F.We expand some properties of ergodic,strong-mixing and weak-mixing to F-ergodic,F-strong-mixing and F-weak-mixing. At the basis of the artical [3],we also introduce the notions of F-2-mixing,F-uniform mixing,.F-Cesaro uniform mixing, F-sweep out,F-double sweep out, F-total sweep out and F-whole sweep out, Moreover,we discuss their properties ,judging and relationship with chaos.The paper is divided into four parts:In chapter one,we mainly introduce the conceptions that will be used in this paper and main conceptions we get.In chapter two, First we give the relations between ergodic,strong-mixing ,weak-mixing and corresponding F-ergodic.Second,we discuss the properties of F-ergodic and relationship with F-sweep out.Finally,we discuss the relationship among F-sweep out ,F-transitive and ergodic.In chapter three,we discuss the properties of F-double sweep out, F-total sweep out and F-whole sweep out and their relationship with chaos.In chapter four.we mainly discuss the properties of F-weak-mixing,F-strong-mixing and F-Cesaro uniform mixing.Furthermore we discuss the description of F-ergodic and F-mixing with respect to the invariant measure for continuous transformation.