On the dimensions of Cantor Julia sets of rational maps
type: Item
instance of: scholarly article
title: On the dimensions of Cantor Julia sets of rational maps
zbMATH Open document ID: 1278.37042
author: Yu Zhai
published in: Journal of Mathematical Analysis and Applications
publication date: 2013-04-22T00:00:00Z
review text: Let \(f\) be a rational map of degree \(d \geq 2\), and denote by \(f^n\) the \(n\)-th iterate of \(f\). The Fatou set \(\mathcal{F}(f)\) is the maximal open set in which the sequence \((f^n)\) is a normal family in the sense of Montel, while the complement of \(\mathcal{F}(f)\) in \(\widehat{\mathbb C}\) is the Julia set \(\mathcal{J}(f)\). A probability measure \(\mu\) supported on \(\mathcal{J}(f)\) is called a conformal measure with exponent \(\alpha\) (\(0 < \alpha \leq 2\)) for \(f\) if the equation \[ \mu(f(A)) = \int_A |f'(z)|^\alpha\,d\mu \] holds for every Borel set \(A \subset \mathcal{J}(f)\) such that \(f_{|A}\) is injective. The existence of a conformal measure for a rational map was proved by \textit{D.~Sullivan} [Lect. Notes Math. 1007, 725--752 (1983; Zbl 0524.58024)]. All the conformal measures of \(f\) form a closed set in the sense of weak convergence. So there exists a conformal measure whose exponent is the infimum of all the exponents of conformal measures of \(f\). This minimal exponent is called the conformal dimension of \(\mathcal{J}(f)\) and is denoted by \(\alpha_*(f)\). Sullivan also proved that for any hyperbolic rational map \(f\) there is a unique conformal measure \(\mu\) supported on \(\mathcal{J}(f)\), whose exponent \(\delta\) equals the Hausdorff dimension of \(\mathcal{J}(f)\). This implies that the conformal dimension of \(\mathcal{J}(f)\) equals the Hausdorff dimension of \(\mathcal{J}(f)\) in the expanding case. Therefore, it is natural to ask whether this also holds for a general rational map. This problem has been studied for several classes of rational maps in the past. In this paper the author proves that \(\alpha_*(f) = \dim_H(\mathcal{J}(f))\) for every rational map \(f\) with a Cantor Julia set satisfying the following two properties: (1) \(\mathcal{F}(f)\) is an attracting domain, that means that \(\mathcal{F}(f)\) contains an attracting fixed point of \(f\). (2) \(\mathcal{J}(f)\) contains no persistently recurrent …
Mathematics Subject Classification ID: 37F10
Mathematics Subject Classification ID: 37F35
Mathematics Subject Classification ID: 37F15
zbMATH DE Number: 6156167
zbMATH Keywords: conformal dimension
zbMATH Keywords: conformal measure
zbMATH Keywords: Hausdorff dimension
zbMATH Keywords: hyperbolic dimension
zbMATH Keywords: rational map
MaRDI profile type: Publication
full work available at URL: j.jmaa.2013.01.052
OpenAlex ID: W2007860237
Wikidata QID: Q121770749
cites work: Hausdorff dimension and conformal measures of Feigenbaum Julia sets
cites work: Julia and John
cites work: On the Existence of Conformal Measures
cites work: On Sullivan's conformal measures for rational maps of the Riemann sphere
cites work: Q3714302
cites work: Rigidity and Expansion for Rational Maps
cites work: Rigidity for real polynomials
cites work: Dimensions of the Julia sets of rational maps with the backward contraction property
cites work: Dynamics of quadratic polynomials. I, II
cites work: Dynamics in one complex variable. Introductory lectures.
cites work: Iterations of holomorphic Collet-Eckmann maps: conformal and invariant measures. Appendix: On non-renormalizable quadratic polynomials
cites work: Lyapunov Characteristic Exponents are Nonnegative
cites work: Proof of the Branner-Hubbard conjecture on Cantor Julia sets
cites work: On the measurable dynamics of real rational functions
cites work: The Hausdorff dimension of the boundary of the Mandelbrot set and Julia sets
cites work: Q4810407
cites work: Q3036339
cites work: Measures and dimensions in conformal dynamics
cites work: Hausdorff and Conformal Measures on Julia Sets with a Rationally Indifferent Periodic Point
cites work: Rational functions with no recurrent critical points
cites work: Q4810408
cites work: No invariant line fields on Cantor Julia sets
DOI: 10.1016/J.JMAA.2013.01.052
description: scientific article; zbMATH DE number 6156167