Non-wandering sets of the powers of maps of a star (Q1811369)

Let \(T\) be a star and \(\Omega(f)\) be the set of non-wandering points of a continuous map \(f: T\to T\). For two distinct prime numbers \(p\) and \(q\) the following theorem is proved: (1) \(\Omega(f^p)\cup \Omega(f^q)= \Omega(f)\) for each \(f\in C(T, T)\) if and only if \(pq> \text{End}(T)\), (2) \(\Omega(f^p)\cap \Omega(f^q)= \Omega(f^{pq})\) for each \(f\in C(T, T)\) if and only if \(p+ q\geq \text{End}(T)\), where \(\text{End}(T)\) is the number of the ends of \(T\).