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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: [email protected]
Mingde Xie
Affiliation:
Department of Mathematics, Guangxi University, Nanning, Guangxi, 530004, People's Republic of China, e-mail: [email protected]
Jinfeng Zhao
Affiliation:
Department of Mathematics, Guangxi University, Nanning, Guangxi, 530004, People's Republic of China, e-mail: [email protected]
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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

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