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September  2003, 9(5): 1175-1184. doi: 10.3934/dcds.2003.9.1175

Non-wandering sets of the powers of maps of a star

Song Shao 1, and  Xiangdong Ye 1,

1. 

Department of Mathematics, University of Science and Technology of China, Hefei, Anhui, 230026, China, China

Received  October 2001 Revised  January 2003 Published  June 2003

Let $T$ be a star and $\Omega(f)$ be the set of non-wandering points of a continuous map $f:T\rightarrow T$. For two distinct prime numbers $p$ and $q$, we prove: (1) $\Omega(f^p)\cup \Omega(f^q)=\Omega(f)$ for each $f \in C(T,T)$ if and only if $pq > End(T)$, (2) $\Omega(f^p)\cap \Omega(f^q)=\Omega(f^{p q})$ for each $f\in C(T,T)$ if and only if $p+q \ge End(T)$, where $End(T)$ is the number of the ends of $T$. Using (1)-(2) and the results in [3], we obtain a complete description of non-wandering sets of the powers of maps of 3-star and 4-star.
Keywords: Non-wandering set, star., tree.
Mathematics Subject Classification: 54H2.
Citation: Song Shao, Xiangdong Ye. Non-wandering sets of the powers of maps of a star. Discrete & Continuous Dynamical Systems, 2003, 9 (5) : 1175-1184. doi: 10.3934/dcds.2003.9.1175
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