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

Song Shao; Xiangdong Ye

doi:10.3934/dcds.2003.9.1175

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2003, Volume 9, Issue 5: 1175-1184. Doi: 10.3934/dcds.2003.9.1175

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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

Received: September 30, 2001

Revised: December 31, 2002

Published: August 2003

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Abstract

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,
tree,
star.

Mathematics Subject Classification: 54H20.

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