Graph maps whose periodic points form a closed set

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:371

Article

Mai, Jie-Hua²  Shao, Song¹ 

[1] Univ Sci & Technol China, Dept Math, Hefei 230026, Anhui, Peoples R China

[2] Shantou Univ, Inst Math, Shantou 515063, Guangdong, Peoples R China

关键词: Graph map;    Periodic point;    omega-limit point;    Recurrent point;   

DOI  :  10.1016/j.jmaa.2010.05.062

来源: Elsevier

【 摘 要 】

A continuous map f from a graph G to itself is called a giaph map. Denote by P(f), R(f), omega(f), Omega(f) and CR(f) the sets of periodic points, recurrent points, omega-limit points, non-wandering points and chain recurrent points of f respectively. It is well known that P(f) subset of R(f) subset of w(f) subset of Omega(f) subset of CR(f). Block and Franke (1983) [5] Proved that if f . 1 -> I is an interval map and P(f) is a closed set, then CR(f) = P(f). In this paper we show that if f . G -> G is a graph map and P(f) is a closed set, then omega(f) = R(f). We also give an example to show that, for general graph maps f with P(f) being a closed set, the conclusion omega(f) = R(f) cannot be strengthened to Omega(f) = R(f) or omega(f)= P(f) (C) 2010 Elsevier Inc All rights reserved.

【 授权许可】

Free