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 ...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）.展开更多
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...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.展开更多
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 cy...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.展开更多
基金Supported by NSFC(Grant Nos.11461003,11261005)NSF of Guangxi(Grant No.2014GXNSFBA118003)
文摘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）.
基金Supported by NNSF of China(Grant No.11461003)SF of Guangxi Univresity of Finance and Economics(Grant Nos.2016KY15,2016ZDKT06 and 2016TJYB06)
文摘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.
文摘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.
