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Equicontinuity of a graph map

Published online by Cambridge University Press:  17 April 2009

Taixiang Sun ,
Yongping Zhang  and
Xiaoyan Zhang
Taixiang Sun
Affiliation:
Department of Mathematics, Guangxi University, Nanning, Guangxi 530004, People's Republic of China, e-mail: [email protected]
Yongping Zhang
Affiliation:
Department of Mathematics, Guangxi University, Nanning, Guangxi 530004, People's Republic of China, e-mail: [email protected]
Xiaoyan Zhang
Affiliation:
Department of Mathematics, Guangxi University, Nanning, Guangxi 530004, People's Republic of China, e-mail: [email protected]
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Let G be a graph, and f: G → G be a continuous map with periodic points. In this paper we show that the following five statements are equivalent.

(1) f is equicontinuous.

(2) There exists some positive integer N such that fN is uniformly convergent.

(3) f is S-equicontinuous for some positive integer sequence S = {n1 < n2 < …}.

(4) Ω(x, f) = ω(x, f) for every x ∈ G.

(5) is a periodic map.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 71 , Issue 1 , February 2005 , pp. 61 - 67
Copyright
Copyright © Australian Mathematical Society 2005

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