On the Lebesgue measure of the Feigenbaum Julia set

TL;DR

This paper proves that the Julia set of the Feigenbaum polynomial has zero Lebesgue measure by showing its Hausdorff dimension is less than 2, resolving a long-standing open problem.

Contribution

It establishes that the Feigenbaum Julia set has zero Lebesgue measure, a significant advancement in understanding its geometric properties.

Findings

Hausdorff dimension of the Feigenbaum Julia set is less than 2

Feigenbaum Julia set has zero Lebesgue measure

Solved a long-standing open question in complex dynamics

Abstract

We show that the Julia set of the Feigenbaum polynomial has Hausdorff dimension less than~2 (and consequently it has zero Lebesgue measure). This solves a long-standing open question.