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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.79.

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Hausdorff dimension and conformal measures of Feigenbaum Julia sets
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by Artur Avila and Mikhail Lyubich PDF
J. Amer. Math. Soc. 21 (2008), 305-363 Request permission

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 $\delta _{\mathrm {cr}}$ is equal to the hyperbolic dimension $\mathrm {HD}_{\mathrm {hyp}}(J(f))$. 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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Additional Information
  • 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
  • Email: artur@ccr.jussieu.fr
  • 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
  • MR Author ID: 189401
  • Email: misha@math.toronto.edu, mlyubich@math.sunysb.edu
  • Received by editor(s): September 20, 2004
  • Published electronically: November 29, 2007
  • © Copyright 2007 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: J. Amer. Math. Soc. 21 (2008), 305-363
  • MSC (2000): Primary 37F25; Secondary 37F35
  • DOI: https://doi.org/10.1090/S0894-0347-07-00583-8
  • MathSciNet review: 2373353