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

Abstract

Let $g(z)={\int }_0^{z}p(t)\exp (q(t))dt+c$ 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 $g^{\prime \prime }$. In order to prove our result, we establish general conditions ensuring that Julia sets have Lebesgue measure zero.

44 pages, 5 figures