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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: stxhql@gxu.edu.cn
Yongping Zhang
Affiliation:
Department of Mathematics, Guangxi University, Nanning, Guangxi 530004, People's Republic of China, e-mail: stxhql@gxu.edu.cn
Xiaoyan Zhang
Affiliation:
Department of Mathematics, Guangxi University, Nanning, Guangxi 530004, People's Republic of China, e-mail: stxhql@gxu.edu.cn
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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

References

[1]Akin, E., Auslander, J. and Berg, K., ‘When is transitive map chaotic?’, in Convergence in Ergodic Theory and Probability (Walter de Gruyter and Co., Berlin, New York, 1996), pp. 25–40.CrossRefGoogle Scholar
[2]Alseda, L. and Moreno, J.M., ‘Linear orderings and the full periodicity kernel for the n-star’, J. Math. Anal. Appl. 180 (1993), 599–616.CrossRefGoogle Scholar
[3]Blanchard, F., Host, B. and Maass, A., ‘Topological complexity’, Ergodic Theory Dynamical Systems 20 (2000), 641–662.CrossRefGoogle Scholar
[4]Blokh, A.M., ‘The set of all iterates is nowhere dense in C ([0, 1], [0, 1])’, Trans. Amer. Math. Soc. 333 (992), 787–798.Google Scholar
[5]Boyce, W., ‘Γ—compact maps on an interval and fixed points’, Trans. Amer. Math. Soc. 160 (1971), 87–102.Google Scholar
[6]Bruckner, A.M. and Hu, T., ‘Equicontinuity of iterates of an interval map’, Tamkang J. Math. 21 (1990), 287–294.CrossRefGoogle Scholar
[7]Cano, J., ‘Common fixed points for a class of commuting mappings on an interval’, Trans. Amer. Math. Soc. 86 (1982), 336–338.Google Scholar
[8]Gu, R., ‘Equicontinuity of maps on figure-eight space’, Southeast Asian Bull. Math. 25 (2001), 413–419.CrossRefGoogle Scholar
[9]Mai, J., ‘The structure of equicontinuous maps’, Trans. Amer. Math. Soc. 355 (2003), 4125–4136.CrossRefGoogle Scholar
[10]Nadler, S.B., Continuum theory, Pure and Appl. Math. 158 (Marcle Dekker, Inc., New York, 1992).Google Scholar
[11]Sun, T., ‘Equicontinuity of σ-maps’, Pure Appl. Math. (Xi'an) 16 (2000), 9–14.Google Scholar
[12]Sun, T., ‘Equicontinuity of tree maps’, J. Math. Study 34 (2001), 125–130.Google Scholar
[13]Valaristos, A., ‘Equicontinuity of iterates of circle maps’, Internat. J. Math. Math. Sci. 21 (1998), 453–458.CrossRefGoogle Scholar