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

TL;DR

This paper proves that for a class of Newton maps derived from certain polynomial-based functions, the Julia sets have Lebesgue measure zero, implying almost everywhere convergence to zeros under specific conditions.

Contribution

It establishes general conditions under which Julia sets of Newton maps have Lebesgue measure zero, extending previous results and linking to convergence properties.

Findings

Julia sets of the considered Newton maps have Lebesgue measure zero

Almost everywhere convergence of iterates to zeros of g under certain conditions

General conditions for Julia sets to have measure zero

Abstract

Let $g(z)=\int_0^zp(t)\exp(q(t))\,dt+c$ where $p,q$ are polynomials and $c\in\mathbb{C}$, 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 $f^n(z)$ converges to zeros of $g$ almost everywhere in $\mathbb{C}$ 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.