Lebesgue measure of Feigenbaum Julia sets

• 成果类型: Article
• 署名作者: Avila, Artur; Lyubich, Mikhail
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2022.195.1.1
• 发表日期: 2022
• 页码: 1-88
• 关键词: hausdorff dimension; quadratic polynomials; complex bounds; typical orbits; siegel disks; RENORMALIZATION; circle; UNIVERSALITY; mappings; DYNAMICS

摘要

We construct Feigenbaum quadratic-like maps with a Julia set of positive Lebesgue measure. Indeed, in the quadratic family Pc : z -> z2 + 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.