Lebesgue measure of Feigenbaum Julia sets

Artur Avila; Mikhail Lyubich

doi:10.4007/annals.2022.195.1.1

Lebesgue measure of Feigenbaum Julia sets

Artur Avila
, Mikhail Lyubich

Mathematical Science Institute

Stony Brook University

University of Zurich
Instituto National de Matemática Pura e Aplicada

Research output: Contribution to journal › Article › peer-review

14 Scopus citations

Abstract

We construct Feigenbaum quadratic-like maps with a Julia set of positive Lebesgue measure. Indeed, in the quadratic family (Formula Presented) 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.

Original language	English
Pages (from-to)	1-88
Number of pages	88
Journal	Annals of Mathematics
Volume	195
Issue number	1
DOIs	https://doi.org/10.4007/annals.2022.195.1.1
State	Published - Jan 2022

Keywords

Feigenbaum map
Julia set
Positive area
Renormalization

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10.4007/annals.2022.195.1.1

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abstract = "We construct Feigenbaum quadratic-like maps with a Julia set of positive Lebesgue measure. Indeed, in the quadratic family (Formula Presented) 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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AB - We construct Feigenbaum quadratic-like maps with a Julia set of positive Lebesgue measure. Indeed, in the quadratic family (Formula Presented) 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.

KW - Feigenbaum map

KW - Julia set

KW - Positive area

KW - Renormalization

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DO - 10.4007/annals.2022.195.1.1

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JO - Annals of Mathematics

JF - Annals of Mathematics

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Lebesgue measure of Feigenbaum Julia sets

Abstract

Keywords

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