Harmonic measure and polynomial Julia sets
type: Item
instance of: scholarly article
title: Harmonic measure and polynomial Julia sets
zbMATH Open document ID: 1036.30017
DOI: 10.1215/S0012-7094-03-11725-1
published in: Duke Mathematical Journal
publication date: 2003-01-01T00:00:00Z
review text: Let \(\Omega\) be a domain in the Riemann sphere \(\widehat{\mathbb C}\), let \(E\subset \widehat{\mathbb C}\), and let \(a\in \Omega\). Let \(\omega(a,E,\Omega)\) denote the harmonic measure in \(\Omega\) of the set \(E\) evaluated at \(a\). An important theorem of \textit{P. W. Jones} and \textit{T. H. Wolff} [Acta Math. 161, 131--144 (1988; Zbl 0667.30020 )] asserts that the Hausdorff dimension of the support of \(\omega\) is at most \(1\). In the paper under review the authors study stronger estimates involving the dimension of harmonic measure. Let \[ \alpha_{a,\Omega}(z)=\liminf_{\delta\to 0}\frac{\log \omega(a,\{w:| w-z| \leq \delta\},\Omega)}{\log \delta}. \] The universal dimension spectrum of harmonic measure is the function \[ \Phi(\alpha)=\sup \text{dim} \{z:\alpha_{a,\Omega}(z)\leq \alpha\}, \] where dim denotes Hausdorff dimension and the supremum is taken over all planar domains \(\Omega\) and all evaluation points \(a\in \Omega\). Let \(\Phi_{\text{sc}}(\alpha)\) be the corresponding quantity with the supremum taken over all simply connected domains. The authors conjecture that for \(\alpha\geq 1\), \(\Phi(\alpha)=\Phi_{\text{sc}}(\alpha)\). They propose the application of methods of Complex Dynamics for the study of the conjecture. The main result of the paper is that the conjecture is true when \(\partial \Omega\) is the Julia set of a polynomial such that the orbits of all critical points escape to \(\infty\). The proof of this deep theorem uses earlier results of \textit{N. G. Makarov} [Algebra Anal. 10, 1--62 (1998); translation in St. Petersbg. Math. J. 10, 217--268 (1999; Zbl 0909.30016)] (involving the pressure function of a polynomial and the universal integral means spectrum), and a construction of \textit{B. Branner} and \textit{J. H. Hubbard} [Acta Math. 160, 143--206 (1988; Zbl 0668.30008)]. Recently I. Binder and P. Jones proved the conjecture mentioned above.
reviewed by: Dimitrios Betsakos
Mathematics Subject Classification ID: 30C85
Mathematics Subject Classification ID: 37F10
Mathematics Subject Classification ID: 37F35
zbMATH DE Number: 1933168
zbMATH Keywords: harmonic measure
zbMATH Keywords: Hausdorff dimension
zbMATH Keywords: Julia set
zbMATH Keywords: conformal Cantor set
Wikidata QID: Q101118435
author: Ilia Binder
author: Nikolai G. Makarov
author: Stanislav Smirnov
MaRDI profile type: Publication
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full work available at URL: s0012-7094-03-11725-1
OpenAlex ID: W2043866337
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description: scientific article; zbMATH DE number 1933168