Fix-points of meromorphic functions and iterated entire functions
type: Item
instance of: scholarly article
title: Fix-points of meromorphic functions and iterated entire functions
zbMATH Open document ID: 0717.30023
DOI: 10.1016/0022-247X(90)90256-F
author: Walter Bergweiler
published in: Journal of Mathematical Analysis and Applications
publication date: 1990-01-01T00:00:00Z
review text: 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 (*).
reviewed by: Norbert Steinmetz
Mathematics Subject Classification ID: 30D05
Mathematics Subject Classification ID: 30D35
zbMATH DE Number: 4181709
zbMATH Keywords: fix-points of meromorphic functions
zbMATH Keywords: iterated entire functions
zbMATH Keywords: Julia set
MaRDI profile type: Publication
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full work available at URL: 0022-247x(90)90256-f
OpenAlex ID: W2030179231
description: scientific article; zbMATH DE number 4181709