Polynomial maps with a Julia set of positive measure

TL;DR

This paper demonstrates the existence of polynomial maps with even degree whose Julia sets have positive Lebesgue measure, addressing a long-standing open problem in complex dynamics.

Contribution

It proves that for sufficiently large even degrees, there are specific parameters making the Julia set of the polynomial have positive measure, solving an old open problem.

Findings

Existence of polynomials with positive measure Julia sets for large even degrees

Addresses a long-standing open problem in complex dynamics

Provides conditions on polynomial degree and parameters

Abstract

In this paper we shall show that there exists L_0 such that for each even integer L >= L_0 there exists $c_1 \in \rz$ for which the Julia set of $z --> z^L + c_1$ has positive Lebesgue measure. This solves an old problem. Editor's note: In 1997, it was shown by Xavier Buff that there was a serious flaw in the argument, leaving a gap in the proof. Currently (1999), the question of polynomials with a positive measure Julia sets remains open.