Lebesgue measure of Feigenbaum Julia sets

TL;DR

This paper constructs specific Feigenbaum quadratic polynomials with Julia sets of positive Lebesgue measure, highlighting differences between hyperbolic and Hausdorff dimensions and revealing a parameter set with positive Hausdorff dimension.

Contribution

It provides the first examples of rational maps with Julia sets of positive Lebesgue measure and distinct hyperbolic and Hausdorff dimensions.

Findings

Julia sets of certain Feigenbaum quadratic polynomials have positive Lebesgue measure

Hyperbolic dimension differs from Hausdorff dimension in these examples

Parameter sets for these maps have positive Hausdorff dimension

Abstract

We construct Feigenbaum quadratic polynomials whose Julia sets have positive Lebesgue measure. They provide first examples of rational maps for which the hyperbolic dimension is different from the Hausdorff dimension of the Julia set. The corresponding set of parameters has positive Hausdorff dimension.