Matings and Newton's method in dimension 2

Abstract: Let p and q be two polynomials of degree d, with connected and locally connected filled in Julia sets Kp and Kq. Let γp, γq: R/Z --> Jp be the Caratheodory loop. We can contruct a topological space Xp,q by taking the disjoint union Kp U Kq and identifying γp(t) with γq(-t); the polynomial p on Kp and q on Kq then fit together to give a map fp,q: Xp,q --> Xp,q, called the mating of p and q.

Sometimes, in fact often, Xp,q is homeomorphic to a sphere, and a homeomorphism can be chosen so that fp,q is a rational function. This is especially exciting when Kp and Kq have empty interiors, in that case γp induces a map R/Z --> \overline C that is a Peano curve.

Matings have been studied extensively, by Douady and Hubbard, Tan Lei, Mary Rees, Shishikura, Milnor, Jiaqi Luo among others.

Matings appear to be a 1-dimensional phenomenon. Recently (with Sebastien Krief and Peter papadopol) we have found that a variant of matings naturally appears in Newton's method in two variables, as applied to the intersection of a pair of parallel lines with a conic.

I my lecture I will go over matings, illustrationg them with the computer, and will explain how they arise in Newton's method.

John Hubbard

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