On the Fatou components of two permutable transcendental entire functions
type: Item
instance of: scholarly article
title: On the Fatou components of two permutable transcendental entire functions
zbMATH Open document ID: 1024.30013
DOI: 10.1016/S0022-247X(02)00664-9
author: Chungchun Yang
author: Xiaoling Wang
published in: Journal of Mathematical Analysis and Applications
publication date: 2003-05-25T00:00:00Z
review text: Two entire functions \(f\) and \(g\) are called permutable if \(f\circ g=g\circ f\). It is an open question whether permutable entire functions must have the same Julia set. (For rational functions this is a classical result of Fatou and Julia.) \textit{I. N. Baker} [Proc. Lond. Math. Soc., III. Ser. 49, 563-576 (1984; Zbl 0523.30017)] showed that this is the case if \(f=g+b\) for some constant \(b\). This was generalized by \textit{K. K. Poon} and \textit{C.-C. Yang} [Ann. Pol. Math. 68, 159-163 (1998; Zbl 0896.30021)] who showed that the Julia sets of permutable entire functions \(f\) and \(g\) are equal if \(f=ag+b\) for constants \(a,b\). Here this result is further generalized. It is shown that it suffices to assume that \(q(f)=aq(g)+b\) for constants \(a,b\) and a non-constant polynomial \(q\). Analogous results are obtained not only for the Julia sets, but also for sets where the iterates tend to \(\infty\) with a certain speed. Further results address the question when the Fatou components of \(f\) and \(g\) are of the same type according to the classification of Fatou components into wandering domains and (preimages of) Böttcher domains, Schröder domains, Landau domains, Siegel discs and Baker domains.
Mathematics Subject Classification ID: 30D05
Mathematics Subject Classification ID: 30D20
Mathematics Subject Classification ID: 37F10
Mathematics Subject Classification ID: 30D35
zbMATH DE Number: 1915724
zbMATH Keywords: Julia set
zbMATH Keywords: Fatou set
zbMATH Keywords: permutable
zbMATH Keywords: entire function
reviewed by: Walter Bergweiler
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1915724