Graph maps whose periodic points form a closed set |
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Authors: | Jie-Hua Mai |
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Affiliation: | a Institute of Mathematics, Shantou University, Shantou, Guangdong, 515063, PR China b Department of Mathematics, University of Science and Technology of China, Hefei, Anhui, 230026, PR China |
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Abstract: | A continuous map f from a graph G to itself is called a graph map. Denote by P(f), R(f), ω(f), Ω(f) and CR(f) the sets of periodic points, recurrent points, ω-limit points, non-wandering points and chain recurrent points of f respectively. It is well known that P(f)⊂R(f)⊂ω(f)⊂Ω(f)⊂CR(f). Block and Franke (1983) [5] proved that if f:I→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 ω(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 ω(f)=R(f) cannot be strengthened to Ω(f)=R(f) or ω(f)=P(f). |
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Keywords: | Graph map Periodic point ω-limit point Recurrent point |
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