Fix-points of meromorphic functions and iterated entire functions
Let f be an entire transcendental function and let \((f_ k)\) denote the sequence of its iterates \((f_ 1=f\), \(f_{k+1}=f\circ f_ k)\). A periodic point \(z_ 0\) of period k is a fix-point of \(f_ k\), but not of \(f_ j\) \((1\leq j<k)\). It is called repelling if \(| f'_ k(z_ 0)| >1\). \textit{I. N. Baker} [Math. Z. 104, 252-256 (1968; Zbl 0172.095)] proved that the Julia set of f is the closure of the repelling periodic points. In the present paper the following is proven: If \[ (*)\quad \frac{| f'(z)|}{1+| f(z)|^ 2}\geq | z|^{\epsilon -1} \] for some \(\epsilon >0\) and some sequence \(z=w_ n\to \infty\) satisfying \[ \limsup| \frac{w_{n+1}}{w_ n}| <\infty, \] then there are infinitely many periodic points of order \(k=2,3,4,... \). This follows from a more general result on the fix- points of \(f\circ g\), where f is transcendental meromorphic satisfying (*), and g is transcendental entire. The author gives various conditions on f to satisfy (*).
scientific article; zbMATH DE number 4181709
scientific article