A class of Newton maps with Julia sets of Lebesgue measure zero

2021/01/20 by Mareike Wolff, Wolff, Mareike · 2 citations
Mathematics · #37F10 (Primary) 30D05 (Secondary) #Complex Variables (math.CV) #Dynamical Systems (math.DS) #FOS: Mathematics #Functional Equations Stability Results #Iterative Methods for Nonlinear Equations #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.2101.08045

openalex publication_date 2021/01/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let g(z)=∫₀^zp(t)exp(q(t)) dt+c where p,q are polynomials and c∈ℂ, and let f be the function from Newton's method for g. We show that under suitable assumptions the Julia set of f has Lebesgue measure zero. Together with a theorem by Bergweiler, our result implies that fⁿ(z) converges to zeros of g almost everywhere in ℂ if this is the case for each zero of g''. In order to prove our result, we establish general conditions ensuring that Julia sets have Lebesgue measure zero.

Cited by

• From pathological to paradigmatic: A retrospective on Eremenko and Lyubich's entire functions
• The escaping set in transcendental dynamics

Related

• Polynomial maps with a Julia set of positive measure
• Newton's method applied to rational functions: Fixed points and Julia sets
• Julia sets of Newton maps of real quadratic polynomial maps on the plane
• On the Lebesgue measure of the Julia set of a quadratic polynomial
• Quadratic and cubic Newton maps of rational functions