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.

Authors

Artur Avila

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

Mikhail Lyubich

Institute for Mathematical Sciences, Stony Brook University, Stony Brook, NY 11794-3660, USA