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  and q  are polynomials and c C . Let f  be the function derived from Newton's method for g . We demonstrate that, under suitable assumptions on the zeros of g , the Julia set of f  has Lebesgue measure zero. Building on a theorem by Bergweiler, our results imply that f n ( z )  converges to the zeros of g  almost everywhere in C  provided this is true for each zero of g  that is not a zero of g . To establish these conclusions, we provide general conditions that guarantee the Julia sets of f  have Lebesgue measure zero.

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