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Equivalent conditions of a tree map with zero topological entropy

Published online by Cambridge University Press:  17 April 2009

Taixiang Sun ,
Mingde Xie  and
Jinfeng Zhao
Taixiang Sun
Affiliation:
Department of Mathematics, Guangxi University, Nanning, Guangxi, 530004, People's Republic of China, e-mail: stxhql@gxu.edu.cn
Mingde Xie
Affiliation:
Department of Mathematics, Guangxi University, Nanning, Guangxi, 530004, People's Republic of China, e-mail: stxhql@gxu.edu.cn
Jinfeng Zhao
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 f: T → T be a tree map with n end-points, SAP(f) the set of strongly almost periodic points of f and CR(f) the set of chain recurrent points of f. Write E(f,T) = {x: there exists a sequence {ki} with 2 ≤ ki ≤ n such that and g = f\CR(f). In this paper, we show that the following three statements are equivalent:

(1) f has zero topological entropy.

(2) SAP(f) ⊂ E(f,T).

(3) Map ωg: x → ω(x,g) is continuous at p for every periodic point p of f.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 73 , Issue 3 , June 2006 , pp. 321 - 327
Copyright
Copyright © Australian Mathematical Society 2006

References

[1]Choi, S., Chu, C. and Lee, K., ‘Recurrence in persistent dynamical systems’, Bull. Austral. Math. Soc. 56 (1997), 467–471.Google Scholar
[2]Li, T. and Ye, X., ‘Chain recurrent points of a tree map’, Bull. Austral. Math. Soc. 59 (1999), 181–186.CrossRefGoogle Scholar
[3]Alseda, L. and Ye, X., ‘No division and the set of periods for tree maps’, Ergodic Theory Dynamical Systems 15 (1995), 221–237.CrossRefGoogle Scholar
[4]Ye, X., ‘No division and the entropy of tree maps’, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 9 (1999), 1859–1865.CrossRefGoogle Scholar
[5]Sun, T., Research of dynamical systems of tree maps, Ph.D. Thesis (in Chinese) (Zhongshan University, 2001).Google Scholar
[6]Sun, T., ‘Chain recurrent points and topological entropy of a tree map‘, Appl. Math. J. Chinese Univ. 17 (2002), 473–478.CrossRefGoogle Scholar
[7]Ye, X., ‘The center and the depth of center of a tree map’, Bull. Austral. Math. Soc. 48 (1993), 347–350.CrossRefGoogle Scholar
[8]Alseda, L., Baldwin, S., Llibre, J. and Misiurewicz, M., ‘Entropy of transitive tree maps’, Topology 36 (1997), 519–532.CrossRefGoogle Scholar
[9]Block, L. and Coppel, W.A., Dynamics in one dimension (Springer-Verlag, New York, 1992).CrossRefGoogle Scholar
[10]Blokh, A.M., ‘Trees with snowflakes and zero entropy maps’, Topology 33 (1994), 379–396.CrossRefGoogle Scholar
[11]Llibre, J. and Misiurewicz, M., ‘Horseshoes, entropy and periods for graph maps’, Topology 32 (1993), 649–664.CrossRefGoogle Scholar