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