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Hausdorff dimension and conformal measures of Feigenbaum Julia sets

Authors: Artur Avila and Mikhail Lyubich
Journal: J. Amer. Math. Soc. 21 (2008), 305-363
MSC (2000): Primary 37F25; Secondary 37F35
Published electronically: November 29, 2007
MathSciNet review: 2373353
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Abstract: We show that contrary to anticipation suggested by the dictionary between rational maps and Kleinian groups and by the ``hairiness phenomenon'', there exist many Feigenbaum Julia sets $ J(f)$ whose Hausdorff dimension is strictly smaller than two. We also prove that for any Feigenbaum Julia set, the Poincaré critical exponent $ \de_{\mathrm{cr}}$ is equal to the hyperbolic dimension $ \HD_{\mathrm{hyp}}(J(f))$. Moreover, if $ \operatorname{area} J(f)=0$, then $ \operatorname{HD}_{\mathrm{hyp}} (J(f))=\operatorname{HD}(J(f))$. In the stationary case, the last statement can be reversed: if $ \operatorname{area} J(f)> 0$, then $ \operatorname{HD}_{\mathrm{hyp}} (J(f))< 2$. We also give a new construction of conformal measures on $ J(f)$ that implies that they exist for any $ \delta\in [\delta_{\mathrm{cr}}, \infty)$, and analyze their scaling and dissipativity/conservativity properties.

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Artur Avila
Affiliation: CNRS UMR 7599, Laboratoire de Probabilités et Modèles aléatoires, Université Pierre et Marie Curie–Boîte courrier 188, 75252–Paris Cedex 05, France

Mikhail Lyubich
Affiliation: Department of Mathematics, University of Toronto, Ontario, Canada M5S 3G3
Address at time of publication: Mathematics Department and IMS, SUNY Stony Brook, Stony Brook, New York 11794

Received by editor(s): September 20, 2004
Published electronically: November 29, 2007
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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