Topological Structure of Non-wandering Set of a Graph Map

Let G be a graph （i.e., a finite one-dimensional polyhedron） and f ： G → G be a continuous map. In this paper, we show that every isolated recurrent point of f is an isolated non-wandering point; every accumulation point of the set of non-wandering points of f with infinite orbit is a two-order accumulation point of the set of recurrent points of f; the derived set of an ω-limit set of f is equal to the derived set of an the set of recurrent points of f; and the two-order derived set of non-wandering set of f is equal to the two-order derived set of the set of recurrent points of f.

Non-Wandering Sets of the Powers of Dendrite Maps

Let （X, d） be a metric space and f be a continuous map from X to X. Denote by EP（f） and Ω（f） the sets of eventually periodic points and non-wandering points of f, respectively. It is well known that for a tree map f, the following statements hold： （1） If x ∈ Ω（f） - Ω（f^n） for some n ≥ 2, then x ∈ EP（f）. （2） Ω（f） is contained in the closure of EP（f）. The aim of this note is to show that the above results do not hold for maps of dendrites D with Card（End（D）） = No （the cardinal number of the set of positive integers）.

Equicontinuity of Maps on a Dendrite with Finite Branch Points

Let （T, d） be a dendrite with finite branch points and f be a continuous map from T to T. Denote by w（x, f） and P（f） the w-limit set of x under f and the set of periodic points of f, respectively. Write Ω（x, f） = {yl there exist a sequence of points xk ∈ T and a sequence of positive integers n1 〈 n2 〈 … such that lim k→∞ Xk = X and lim k→∞ f nk （xk） = y}. In this paper, we show that the following statements are equivalent： （1） f is equicontinuous. （2） w（x, f） = Ω（x, f） for any x ∈ T. （3） ∩ ∞ n=1 f n（T） = P（f）, and w（x, f） is a periodic orbit for every x ∈ T and map h： x → w（x, f） （x ∈ T） is continuous. （4） Ω（x, f） is a periodic orbit for any x ∈ T.

Topological Entropy of a Graph Map

Let G be a graph and f： G → G be a continuous map. Denote by h（f）, P（f）, AP（f）, R（f） and w（x, f） the topological entropy of f, the set of periodic points of f, the set of almost periodic points of f, the set of recurrent points of f and the w-limit set of x under f, respectively. In this paper, we show that the following statements are equivalent： （1） h（f） 〉 O. （2） There exists an x ∈ G such that w（x, f） ∩ P（f） ≠θ and w（x, f） is an infinite set. （3） There exists an x ∈ G such that w（x, f） contains two minimal sets. （4） There exist x, y ∈G such that w（x, f） - w（y, f） is an uncountable set andw（y,f）∩w（x,f）≠θ. （5） There exist anx C Gand a closed subset A w（x,f） with f（A） A such that w（x,f） - A is an uncountable set. （6） R（f） - nP（f） ≠θ. （7） f｜P（f） is not pointwise equicontinuous.

The Depths and the Attracting Centres for Continuous Maps on a Dendrite Whose Rank is Finite

Let D be a dendrite and f be a continuous map on D.Denote by R(f),Ω(f)andω(x,f)the set of recurrent points,the set of non-wandering points and the set ofω-limit points of x under f,respectively.WriteΩ_(k+1)(f)=Ω(f|_(()Ω_(k)(f)))andω^(k+1)(f)=∪_(()x∈ω~k(f))ω(x,f)for any positive integer k,whereΩ_(1)(f)=Ω(f)andω(f)=∪_(x∈D)ω(x,f).ω~m(f)is called the attracting centre of f ifω^(m+1)(f)=ω~m(f).In this paper,we show that if the rank of D is n-1,then we have the following results:(1)ω^(n+2)(f)=ω^(n+1)(f)and the attracting centre of f isω^(n+1)(f);(2)Ω_(n+2)(f)=■and the depth of f is at most n+2.Further,if the set of(n-1)-order accumulation points of Br(D)(the set of branch points of D)is a singleton,thenΩ_(n+1)(f)=■and the depth of f is at most n+1.Besides,we show that there exist a dendrite D_(1)whose rank is n-1 and the set of(n-1)-order accumulation points of Br(D_(1))is a singleton,and a continuous map g on D_(1)such thatω^(n+1)(g)≠ω~n(g)andΩn(f)≠■.

The Strongly Simple Cycles with Given Rotation Pairs of an Interval Map

In this paper, we introduce the notion of the strongly simple cycles with some rotation pair for interval maps and prove that, if an interval map has a cycle with given rotation pair, then it, has a strongly simple cycle with the same rotation pair.

Stability of Periodic Orbits and Return Trajectories of Continuous Multi-valued Maps on Intervals

Let I be a compact interval of real axis R, and （L, H） be the metric space of all nonempty closed subintervals of I with the Hausdorff metric H and f ： I →L be a continuous multi-valued map. Assume that Pn = （x0, x1,..., xn） is a return trajectory of f and that p ∈ [min Pn, max Pn] with p ∈ f（p）. In this paper, we show that if there exist k （≥ 1） centripetal point pairs of f （relative to p） in {（xi;xi＋l） ： 0 ≤ i ≤ n- 1} and n =sk＋r （0 ≤ r ≤ k - 1）, then f has an R-periodic orbit, where R=s＋1 ifsiseven, and R =s if s is odd and r = 0, and R=s＋2 if s is odd and r 〉0. Besides, we also study stability of periodic orbits of continuous multi-valued maps from I to L.