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

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

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.
Citation: Song Shao, Xiangdong Ye. Non-wandering sets of the powers of maps of a star. Discrete & Continuous Dynamical Systems - A, 2003, 9 (5) : 1175-1184. doi: 10.3934/dcds.2003.9.1175
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