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

Abstract

Let g(z)=int _0^zp(t)exp (q(t)),dt+c where p, q are polynomials and cin {mathbb {C}}, and let f be the function from Newton’s method for g. We show that under suitable assumptions on the zeros of g'' 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'' that is not a zero of g or g'. In order to prove our result, we establish general conditions ensuring that Julia sets have Lebesgue measure zero.