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

• 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.
Mathematics Subject Classification: 54H20.

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