Newton maps as matings of cubic polynomials (Q2827963)

Mating two polynomials \(f_1\) and \(f_2\) of the same degree means, roughly speaking, gluing together their filled Julia sets along their boundaries (in reverse order) and getting the resulting set homeomorphic to the sphere with a new map induced by \(f_1\) and \(f_2\) of the sphere to itself [\textit{A. Douady}, Sémin. Bourbaki, 35e année, 1980/81, Exp. 599, Astérisque 105--106, 39--63 (1983; Zbl 0532.30019)]. The authors prove existence and uniqueness of mating for a large class of pairs of cubic polynomials such that one of them has one fixed critical point and another polynomial has two fixed critical points. The resulting map is essentially a Newton map \(N(z)=z-P(z)/P'(z)\), where \(P\) is a cubic polynomial.