On the non-escaping set of \(e^{2\pi i\theta} \sin (z)\)
type: Item
instance of: scholarly article
title: On the non-escaping set of \(e^{2\pi i\theta} \sin (z)\)
zbMATH Open document ID: 1152.37018
author: Gaofei Zhang
published in: Israel Journal of Mathematics
publication date: 2008-09-03T00:00:00Z
review text: Let \(\theta\) be an irrational number of bounded type and let \(f_{\theta}(z)=e^{2\pi i\theta}\sin z\). The author proved in a previous paper that the Siegel disk of \(f_{\theta}\) around 0 is bounded by a quasicircle passing through the critical points \(\pi/2\) and \(-\pi/2\). The main result of this paper is that the set of all \(z\) in the Julia set of \(f_{\theta}\) which do not tend to \(\infty\) under iteration has measure zero. In contrast, by a result of \textit{C. McMullen} [Trans. Am. Math Soc. 300, 329-342 (1987; Zbl 0618.30027)], the set of all \(z\) which tend to \(\infty\) under iteration has positive measure. Besides the methods of his previous paper, the author also uses ideas of \textit{C. L. Petersen} [Acta Math. 177, No. 2, 163-224 (1996; Zbl 0884.30020)] who proved that the Julia set of \(e^{2\pi i\theta}z+z^2\) has measure zero for \(\theta\) of bounded type.
reviewed by: Walter Bergweiler
Mathematics Subject Classification ID: 37F10
Mathematics Subject Classification ID: 30D05
Mathematics Subject Classification ID: 37F50
zbMATH DE Number: 5320460
zbMATH Keywords: Julia set
zbMATH Keywords: Fatou set
zbMATH Keywords: escaping set
zbMATH Keywords: Siegel disk
MaRDI profile type: Publication
full work available at URL: s11856-008-1011-4
OpenAlex ID: W2090641624
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DOI: 10.1007/S11856-008-1011-4
description: scientific article; zbMATH DE number 5320460