Recurrent critical points and typical limit sets for conformal measures
type: Item
instance of: scholarly article
title: Recurrent critical points and typical limit sets for conformal measures
zbMATH Open document ID: 0966.30021
DOI: 10.1016/S0166-8641(99)00141-8
author: Alexander Blokh
author: John C. Mayer
author: Lex Oversteegen
published in: Topology and its Applications
publication date: 2001-08-06T00:00:00Z
review text: Let \(f\) be a rational function. A measure \(\mu\) on the Riemann sphere is called conformal of exponent \(\alpha\) if \(\mu(f(A))=\int_{A}|f'(z)|^{\alpha}d\mu\) for all measurable sets \(A\) where \(f\) is injective. Suppose that \(\mu\) is a conformal measure of exponent \(\alpha\) and that the Julia set \(J(f)\) of \(f\) contains a subset \(L\) of positive \(\mu\)-measure such that the points in \(L\) are not preimages of critical or parabolic points and such that the \(\omega\)-limit sets of the points in \(L\) are not contained in the union of the \(\omega\)-limit sets of the recurrent critical points. It is shown that then \(f\) is exact, ergodic and conservative, that \(\mu\) is unique, non-atomic and supp\((\mu)=J(f)\), and that \(\omega(z)=J(f)\) for \(\mu\)-almost every point \(z\). Moreover, \(\alpha\) is the minimal exponent for which a conformal measure with support on \(J(f)\) exists. This result extends previous results by various other authors, e.g. results by \textit{M. Yu. Lyubich} [Dokl. Akad. Nauk SSSR 268; 29-32 (1983; Zbl 0595.30034)] and \textit{C. T. McMullen} [Complex Dynamics and Renormalization, Princeton, NJ (1994)] dealing with the case that \(\mu\) is the Lebesgue measure.
reviewed by: Walter Bergweiler
Mathematics Subject Classification ID: 30D05
Mathematics Subject Classification ID: 37F10
Mathematics Subject Classification ID: 37A25
zbMATH DE Number: 1545863
MaRDI profile type: Publication
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full work available at URL: s0166-8641(99)00141-8
OpenAlex ID: W2467129064
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description: scientific article; zbMATH DE number 1545863