Regularity and fast escaping points of entire functions
type: Item
instance of: scholarly article
zbMATH Open document ID: 1307.30060
author: Philip Jonathan Rippon
author: Gwyneth Mary Stallard
publication date: 2014-11-12T00:00:00Z
full work available at URL: https://arxiv.org/abs/1301.2193
Mathematics Subject Classification ID: 30D05
Mathematics Subject Classification ID: 37F10
zbMATH DE Number: 6369071
zbMATH Keywords: entire functions
zbMATH Keywords: iteration
zbMATH Keywords: Eremenko-Lyubich class
zbMATH Keywords: escaping set
zbMATH Keywords: Julia set
MaRDI profile type: Publication
OpenAlex ID: W2962721538
arXiv ID: 1301.2193
DOI: 10.1093/IMRN/RNT111
published in: IMRN. International Mathematics Research Notices
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title: Regularity and fast escaping points of entire functions
review text: Let \(f\) be a transcendental entire function \(f\), and denote by \(f^n\), \(n=0,1,2, \dots\), the \(n\)th iterate of \(f\). The escaping set \(I(f)\) of \(f\) is defined as follows NEWLINE\[NEWLINE I(f)=\big\{z \;\big|\; f^n (z)\to \infty \text{ as } n\to \infty\big\}.NEWLINE\]NEWLINE \textit{A. E. Eremenko} [Banach Cent. Publ. 23, 339--345 (1989; Zbl 0692.30021)] proved that all components of \(\overline{I(f)}\) are unbounded and conjectured that all components of \({I(f)}\) itself are unbounded as well. Using the fast escaping set \(A(f)\) introduced by \textit{W. Bergweiler} and \textit{A. Hinkkanen} in [Math. Proc. Camb. Philos. Soc. 126, No. 3, 565--574 (1999; Zbl 0939.30019)], the authors proved in [Proc. Am. Math. Soc. 133, No. 4, 1119--1126 (2005; Zbl 1058.37033)] that \(I(f)\) has at least one unbounded component.NEWLINENEWLINELet \(M(r)=M(r,f)=\max\{ |f(z)|: |z|=r\}\), \(r>0\), and set NEWLINE\[NEWLINE A(f)=\big\{z : \exists l \in \mathbb{N}\; \big |\;f^{n+l}(z)|\geq M^n(R), \;n\in{\mathbb N}\big\},NEWLINE\]NEWLINE where \(M^n(r)\) denotes the iterations of \(M(r)\), and \(R>0\) is any value such that \(M(r)>r\) for \(r\geq R\). Let \(\mu_\varepsilon(r)=\varepsilon M(r)\), \(r>0\), \(0<\varepsilon<1\). Consider the sets NEWLINENEWLINENEWLINE\[NEWLINE Q_\varepsilon(f)=\big\{z : \exists l \in \mathbb{N}\;\big |\;f^{n+l}(z)|\geq \mu_\varepsilon^n(R),\; n\in\mathbb{N}\big\},NEWLINE\]NEWLINE where \(R>0\) is any value such that \(\mu_\varepsilon(r)>r\) for \(r\geq R\). We define the set \(Q(f)=\bigcup_{0<\varepsilon<1} Q_\varepsilon(f)\) to be the quite fast escaping set of \(f\). One has NEWLINE\[NEWLINE A(f)\subset Q_\varepsilon(f)\subset Q(f) \subset I(f), \quad 0<\varepsilon<1.NEWLINE\]NEWLINE The authors study the relations between \(A(f)\) and \(Q(f)\).NEWLINENEWLINE{Theorem 1.1.} Let \(f\) be a transcendental entire function in the Eremenko-Lubich class \(\mathcal{B}\). Then \(Q(f)=A(f)\).NEWLINENEWLINERecall that the class \(\mathcal{B}\) consists of …
reviewed by: Igor Chyzhykov
description: scientific article; zbMATH DE number 6369071