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

The paper proves a general zero-measure criterion (Theorem 1.3) and then verifies its three hypotheses for Newton maps of g(z)=∫_0^z p(t)e^{q(t)}dt+c under the stated dynamical assumption on the zeros of g'' (Theorem 1.1). Specifically: (i) it shows P(f)∩J(f) is finite (Lemma 6.2, using that Newton maps have no finite asymptotic values and that each free critical point is attracted to a periodic cycle) ; (ii) it proves uniform thinness near finite postsingular values outside a large disk (Section 11) ; and (iii) it establishes thinness at infinity (end of Section 11) . The candidate solution outlines exactly this route, invoking the same zero-measure criterion (Theorem 1.3) and the sectorial asymptotics for g and f (Corollaries 4.2 and 4.4) , and correctly uses Lemma 6.1 and Lemma 6.2 to control singular values and the postsingular set . The only substantive quibble is a minor slip in the written asymptotic for f: the model omits the e^{-q(z)} factor in the small term (it writes −c_k/p(z) instead of −c_k e^{-q(z)}/p(z)), but this does not affect the logic of applying the criterion. Both arguments are thus materially the same and correct.