On the complement of the Mandelbrot set
type: Item
instance of: scholarly article
title: On the complement of the Mandelbrot set
zbMATH Open document ID: 0817.30012
DOI: 10.1007/BF02937511
published in: Israel Journal of Mathematics
publication date: 1995-07-18T00:00:00Z
review text: The Mandelbrot set \(M= \{c: J_ c\) is connected\}, where \(J_ c\) is the Julia set of \(f_ c(z)= z^ 2+ c\). The complement of \(M\) is conformally equivalent to the exterior of a disc \(D\) by a map \(\Phi\) with \(\Phi(\infty)= \infty\), \(\Phi'(\infty) =1\), \(\Phi(D^ c)= M^ c\). The boundary of \(M\) is extremely intricate and the study of the boundary properties of \(\Phi\) correspondingly difficult. The paper studies this question for certain points of \(\partial M\). A hyperbolic component \(W\) of the interior of \(M\) is a set of \(c\) in which \(f_ c\) has an attracting cycle \(\alpha(c)\) of some fixed period \(m\). Let \(c_ 0\) be a boundary point of \(W\), such that the multiplier \(\lambda(c)\) of \(\alpha(c)\) has the value \(\exp(2\pi i\nu)\), where the rational number \(\nu= p/q\) in lowest terms. Roughly speaking the paper describes the connection between \(\lambda(c)\), the so-called rotation number of the cycle \(\alpha(c)\) and parameters related to the value of the mapping function \(\Phi\) as \(c\) passes from \(W\) to \(M^ c\) through \(c_ 0\). The arguments involve an extension of an inequality of \textit{J. C. Yoccoz} [Sur la taille des membres de l'ensemble de Mandelbrot, Manuscript 1987], and a foliation of \(\widehat{\mathbb{C}}\backslash J_ c\) described as the `hedgehog'.
Mathematics Subject Classification ID: 30D05
Mathematics Subject Classification ID: 37B99
zbMATH DE Number: 720058
zbMATH Keywords: Mandelbrot set
zbMATH Keywords: rotation number
zbMATH Keywords: Julia set
zbMATH Keywords: multiplier
zbMATH Keywords: cycle
author: Genadi Levin
reviewed by: I. N. Baker
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02937511
OpenAlex ID: W1983164255
description: scientific article; zbMATH DE number 720058