Periodic points and normal families (Q2723482)

For a domain \(D \subset \mathbb C\) and a holomorphic function \(f : D \to \mathbb C\), the iterates \(f^n : D_n \to \mathbb C\) are defined by \(D_1:=D\), \(f^1:=f\) and \(D_n:=f^{-1}(D_{n-1})\), \(f^n:=f^{n-1} \circ f\) for \(n \in \mathbb N\), \(n \geq 2\). Then \(D_{n+1} \subset D_n \subset D\). A point \(\zeta \in D\) is called a periodic point of period \(n\) of \(f\) if \(\zeta \in D_n\) and if \(\zeta\) is a fixed point of \(f^n\) but not of \(f^m\) for \(1 \leq m \leq n-1\). There are various results on holomorphic (in particular entire) functions and their periodic points. For example, a theorem of W.~Bergweiler [Complex Variables, Theory Appl. 17, 57-72 (1991; Zbl 0748.30020)] states that a transcendental entire function has infinitely many periodic points of period \(n\) for every \(n \geq 2\). This article is concerned with a normal family analogues of such results. NEWLINENEWLINENEWLINEA family \(\mathcal F\) of holomorphic functions in a domain \(D \subset \mathbb C\) is called quasinormal if for each sequence \((f_k)\) in \(\mathcal F\) there exists a subsequence \((f_{k_j})\) and a finite set \(E \subset D\) such that \((f_{k_j})\) converges locally uniformly in \(D \setminus E\). If the cardinality of \(E\) can be bounded independently of the sequence \((f_k)\), and if \(q\) is the smallest such bound, then \(\mathcal F\) is called quasinormal of order \(q\). Now, for a domain \(D \subset \mathbb C\), \(m \in \mathbb N\), \(m \geq 2\) let \(\mathcal F_m = \mathcal F_m(D)\) be the family of all holomorphic functions \(f : D \to \mathbb C\) for which there exists \(n=n(f) \geq m\) such that \(f\) has no periodic point of period \(n\). There holds \(\mathcal F_{m+1} \subset \mathcal F_m\), and since \(\mathcal F_m\) contains all linear transformations, none of the families \(\mathcal F_m\) is normal. But the main result of this article states that \(\mathcal F_2(D)\) is quasinormal of order \(1\). Also the sequences in \(\mathcal F_2(D)\) without convergent subsequences are characterized. NEWLINENEWLINENEWLINEThe method of proof uses the Ahlfors theory of covering surfaces and some elementary concepts from graph theory. Also this method yields a new proof of the above mentioned theorem of W.~Bergweiler. Finally, repelling period points of holomorphic functions are considered.