Continuity of Julia sets and its Hausdorff dimension of \(P_{c}(z) = z^{d} + c\)
type: Item
instance of: scholarly article
title: Continuity of Julia sets and its Hausdorff dimension of \(P_{c}(z) = z^{d} + c\)
zbMATH Open document ID: 1377.37073
author: Wei Zhuang
published in: Hokkaido Mathematical Journal
publication date: 2013-11-25T00:00:00Z
full work available at URL: https://projecteuclid.org/euclid.hokmj/1384273388
Mathematics Subject Classification ID: 37F10
Mathematics Subject Classification ID: 37F35
zbMATH DE Number: 6232361
zbMATH Keywords: Julia set
zbMATH Keywords: Hausdorff dimension
zbMATH Keywords: net
zbMATH Keywords: conformal measure
Wikidata QID: Q123204713
MaRDI profile type: Publication
OpenAlex ID: W2029079670
DOI: 10.14492/HOKMJ/1384273388
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review text: Let \(R\) be a rational map of degree \(d = \deg{R} \geq 2\) on the complex sphere \(\widehat{\mathbb{C}}\), and denote by \(R^n\) the \(n\)-th iterate of \(R\). The Fatou set \(F(R)\) is the maximal open set in which the sequence \((R^n)\) is a normal family in the sense of Montel, while the complement of \(F(R)\) in \(\widehat{\mathbb{C}}\) is the Julia set \(J(R)\). It is well known that \(J(R)\) is a perfect set, and thus uncountable. If it is disconnected, then it has infinitely many components.NEWLINENEWLINEIf \(\mathcal{C}\) denotes the set of critical points of \(R\) then the set of critical values of \(R^n\) is given by NEWLINE\[NEWLINE Ctv_n(R) = R(\mathcal{C}) \cup R^2(\mathcal{C}) \cup\dotsb\cup R^n(\mathcal{C}). NEWLINE\]NEWLINE Then the \(\omega\)-limit set \(\Omega(R)\) is the set of all \(z \in \widehat{\mathbb{C}}\) such that there exists \(c \in Ctv_n(R)\) and a sequence \((n_k)\) of positive integers such that \(n_k \to \infty\) as \(k\to\infty\) and \(R^{n_k}(c) \to z\) as \(k\to\infty\).NEWLINENEWLINEThe author calls a critical point \(c\) of \(R\) recurrent if \(c \in \Omega(R)\) and non-recurrent otherwise. Such maps are denoted as NCP maps. They are called semi-hyperbolic if the critical points are not recurrent by \(R\) and without parabolic cycles.NEWLINENEWLINEA sequence \((R_n)\) of rational maps converges to \(R\) \textit{algebraically} if \(\deg{R_n}=\deg{R}\) and, when \(R_n\) is expressed as the quotient of two polynomials, the coefficients of which can be chosen such that they converge to those of \(R\). Equivalently, there holds \(R_n \to R\) as \(n\to\infty\) in the spherical metric.NEWLINENEWLINENow, let \(R_n \to R\) as \(n\to\infty\) algebraically, and let \(b \in J(R)\) be a pre-periodic critical point with \(R^j(b)=R^k(b)\) for some \(j>k>0\). Furthermore, suppose that for all such \(b\) and for all sufficiently large \(n\) the maps \(R_n\) have critical points \(b_n \in J(R_n)\) with the same multiplicity as \(b\), \(b_n \to …
description: scientific article; zbMATH DE number 6232361