Topological Limit Of Trajctories Of Sets

In this paper, we will study dynamic systems generated by continuous maps of a compact metric space (X,d) into itself. We mainly discuss topological limit of trajectories of non-empty subsets of X. We assume that A X is a non-empty subset of X, and f C0(X). Denote by S(A, f) and I(A, f) be the upper topological limit and the lower topological limit of the trajectory of A under /, respectively. Let P(f) and F(f) be the set of all periodic points and the set of all fixed points, respectively. Put K(X) = {A X | A is a non-empty closed subset of X}, and Kf{X) = {A K(X)|I(A,f)=}.In the second section, we will present some fundamental properties of S(A, f) and I (A, f), and obtain some equivalent conditions of the existence of the topological limit of A, and so on.In the third section, we study the topological limit of the trajectory of the subset A of a tree and the connectedness of S(A,f) and I(A,f). Our main result is: Let T be a tree, A T be a non-empty connected subset of T, and f G 0(T). If P(f) = F(f), then the topological limit of the trajectory of A exists.In the forth section, we will discuss the continuity of Sf K{X)- K(X) (Sf(A) = S(A,f)) and If: Kf(X) - K(X) (If(A) = I(A,f)), and give some conditions for Sf and If to be continuous.