Lebesgue measure of Feigenbaum Julia sets

Abstract

We construct Feigenbaum quadratic-like maps with a Julia set of positive Lebesgue measure. Indeed, in the quadratic family $P_c: z \mapsto z^2+c$ the corresponding set of parameters $c$ is shown to have positive Hausdorff dimension. Our examples include renormalization fixed points, and the corresponding quadratic polynomials in their stable manifold are the first known rational maps for which the hyperbolic dimension is different from the Hausdorff dimension of the Julia set.

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      title = {Rigidity of critical circle mappings. {II}},
      journal = {J. Amer. Math. Soc.},
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      issn = {0894-0347},
      mrclass = {37E10 (37C15 37E20 37F25)},
      mrnumber = {1711394},
      mrreviewer = {Grzegorz \'{S}wi\polhk atek},
      doi = {10.1090/S0894-0347-99-00324-0},
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      journal = {Sov. Math., Dokl.},
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      volume = {30},
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      mrclass = {30D05 (58F11)},
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      journal = {J. London Math. Soc. (2)},
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      doi = {10.1112/jlms/s2-36.3.458},
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      author = {Lyubich, Mikhail},
      title = {How big is the set of infinitely renormalizable quadratics?},
      booktitle = {Voronezh {W}inter {M}athematical {S}chools},
      series = {Amer. Math. Soc. Transl. Ser. 2},
      volume = {184},
      pages = {131--143},
      publisher = {Amer. Math. Soc., Providence, RI},
      year = {1998},
      mrclass = {37F10 (30D05 37F25 37F45)},
      mrnumber = {1729930},
      mrreviewer = {Hartje Kriete},
      doi = {10.1090/trans2/184/09},
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      }
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      venue = {Zürich},
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      year = {1995},
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      VOLUME = {2},
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      PAGES = {29--44},
      MRCLASS = {58F23},
      MRNUMBER = {1613051},
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      DOI = {10.1090/S1088-4173-98-00019-8},
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Authors

Artur Avila

Institut für Mathematik, Universität Zürich, Winterthurerstrasse 190, CH-8057 Zürich, Switzerland and IMPA, Estrada Dona Castorina 110, Rio de Janeiro, Brazil

Mikhail Lyubich

Department of Mathematics, Stony Brook University, Stony Brook, NY 11794, USA