Polynomial maps with a Julia set of positive measure

1994/02/16 by Tomasz Nowicki, Nowicki, Tomasz, Sebastian van Strien +1
Mathematics · #Advanced Topology and Set Theory #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Meromorphic and Entire Functions

paper · pdf · doi:10.48550/arxiv.math/9402215

openalex publication_date 1994/02/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper we shall show that there exists L₀ such that for each even integer L >= L₀ there exists c₁ ∈ \rz for which the Julia set of z --> z^L + c₁ 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.

Citations

• Quasisymmetric conjugacies between unimodal maps
• Hyperbolicity is dense in the real quadratic family
• Induced expansion for quadratic polynomials

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• Lebesgue measure of Feigenbaum Julia sets
• Transcendental Julia sets of positive finite Lebesgue measure
• On the Lebesgue measure of the Julia set of a quadratic polynomial
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