Equicontinuity of Maps on a Dendrite with Finite Branch Points  

作　　者：Tai Xiang SUN Guang Wang SU Hong Jian XI Xin KONG 

机构地区：[1]College of Information and Statistics,Guangxi University of Finance and Economics [2]College of Mathematics and Information Science,Guangxi University

出　　处：《Acta Mathematica Sinica,English Series》2017年第8期1125-1130,共6页数学学报（英文版）

基　　金：Supported by NNSF of China(Grant No.11461003);SF of Guangxi Univresity of Finance and Economics(Grant Nos.2016KY15,2016ZDKT06 and 2016TJYB06)

摘　　要：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.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.

关 键 词：Dendrite map EQUICONTINUITY periodic point ε-limit set 

分 类 号：O144[理学—数学]