The Set Of Positive Upper Banach Density Recurrence

In the long run of Dynamical System, after transient phenomena have passed away,what remains is recurrence. The present paper studies a particular recurrence, i.e recurrence of positive upper Banach density. Similar to discussion of the set of R(f). we discuss the property of the set of positive upper Banach density recurrence. In the prelace, we give a review about the historical background of Dynamical System and research achivements in these fields. The present paper is divided into three parts.In chapter one, we introduce some fundamental knowledge in topological dynamical system, family F, equicontinuity, sensitive dependence on initial conditions, scattering and measure theory. We also present many well-known consequence and main result in this paperIn chapter two. we discuss the set of positive upper Bananch density recurrence-R+B(f) First, we study equivalent descriptions of a point belonging to it, i.e Theorem 2.1.8. Second, the relationship among R+B(f). the set of support and measure center, .e Theorem 2.2.5, Theorem2.2.7. Finaly, we discuss the property of R+B(f), i.eTheorem 2.3.1, Theorem 2.3.2, Theorem 2.3.6. In the proceeding of the proofs, we generalize some results in [4],[5].In chapter three, using R+B(f), we give some equivalent conditions for E-system. i.e Theorem 3.1.1. When R+B(f) = X( that is the whole space we discussed), we get that some conditions are equivalent, i.e Theorem 3.2.2, Theorem 3.2.6.