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

Let g (z) = ∫ 0 z p (t) exp (q (t)) d t + 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 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 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. [ABSTRACT FROM AUTHOR]