Distortion control of conjugacies between quadratic polynomials
type: Item
instance of: scholarly article
title: Distortion control of conjugacies between quadratic polynomials
zbMATH Open document ID: 1242.37033
author: GuiZhen Cui
author: Lei Tan
published in: Science China. Mathematics
publication date: 2010-05-26T00:00:00Z
review text: The paper is devoted to a theorem on the dynamics of the quadratic polynomial family \(Q_c(z):=z^2+c\) with \(c\in\mathbb C\). For each \(c\), \[ K_c:=\{z\in\mathbb C:(Q_c^{\circ n}(z))_{n\in\mathbb N}\text{ is bounded}\} \] is a compact set called the filled-in Julia set. It is known that this set is connected if and only if \(c\in K_c\), and the set of all such \(c\) is called the Mandelbrot set, which we denote by \(\mathcal M\). Let \(c\in\mathcal M\). To study the boundary of \(K_c\) one employs the conformal (onto) mapping \(\phi_c:\overline{\mathbb C}\setminus K_c\to\{z\in\overline{\mathbb C}:|z|>1\}\) normalized by \(\phi_c(\infty)=\infty\) and \(\phi_c'(\infty)>0\). Given \(\theta\in\mathbb R\), the set \[ R_c(\theta):=\{\phi_c^{-1}(re^{2\pi i\theta}):r>1\} \] is called the dynamic \(\theta\)-ray for \(Q_c\). In a similar way one defines parameter rays. Indeed, it is known that complement \(\overline{\mathbb C}\setminus \mathcal M\) of the Mandelbrot set is a simply connected domain containing~\(\infty\). Let \(\Phi\) be the conformal mapping of this domain onto \(\{z\in\overline{\mathbb C}:|z|>1\}\) normalized by \(\Phi(\infty)=\infty\) and \(\Phi'(\infty)>0\). For \(\theta\in\mathbb R\), the parameter \(\theta\)-ray is defined as the set \[ R_M(\theta):=\{\Phi^{-1}(re^{2\pi i\theta}):r>1\}. \] We say that the dynamic ray \(R_c(\theta)\) (or parameter ray \(R_M\)) lands at a point \(p\) if \(\phi_c^{-1}(re^{2\pi i\theta})\to p\) (or \(\Phi^{-1}(re^{2\pi i\theta})\to p\), respectively) as \(r\to1+0\). The image \(Q_c(R_c(\theta))\) of any dynamic ray \(R_c(\theta)\) is again a dynamic ray, namely~\(R_c(2\theta)\). A dynamic ray~\(R_c(\theta)\) is called periodic if \(Q^{\circ n}_c(R_c(\theta))=R_c(\theta)\) for some \(n\in\mathbb N\), and strictly preperiodic if it is not periodic but \(Q^{\circ m}_c(R_c(\theta))\) is periodic for some \(m\in\mathbb N\). (Clearly, this depends only on \(\theta\), but not on~\(c\).) The following result is due to Douady and …
reviewed by: Pavel Gumenyuk
Mathematics Subject Classification ID: 37F45
Mathematics Subject Classification ID: 30C35
Mathematics Subject Classification ID: 30C55
Mathematics Subject Classification ID: 30D05
Mathematics Subject Classification ID: 37F10
Mathematics Subject Classification ID: 37F20
zbMATH DE Number: 5712103
zbMATH Keywords: quadratic polynomial
zbMATH Keywords: Mandelbrot set
zbMATH Keywords: external ray
zbMATH Keywords: dynamic ray
zbMATH Keywords: parameter ray
zbMATH Keywords: Misiurewicz point
zbMATH Keywords: univalent function
zbMATH Keywords: distortion
zbMATH Keywords: Kraus-Nehari theorem
MaRDI profile type: Publication
full work available at URL: s11425-010-0041-7
OpenAlex ID: W2147171119
cites work: Schwarzian derivatives, the Poincaré metric and the kernel function
cites work: The Poincaré Metric of Plane Domains
cites work: Q3744437
cites work: Frontiers in complex dynamics
cites work: Q4810409
DOI: 10.1007/S11425-010-0041-7
description: scientific article; zbMATH DE number 5712103