**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 *K*_{p} and
*K*_{q}. Let γ_{p},
γ_{q}: **R**/**Z** --> *J*_{p} be the
Caratheodory loop. We can contruct a topological space *X*_{p,q} by
taking the disjoint union *K*_{p} U *K*_{q}
and identifying
γ_{p}(*t*) with
γ_{q}(-*t*); the polynomial *p* on
*K*_{p} and
*q* on *K*_{q} then fit together to give a map
*f*_{p,q}: *X*_{p,q} -->
*X*_{p,q}, called the mating of *p* and *q*.

Sometimes, in fact often, *X*_{p,q}
is homeomorphic to a sphere, and
a homeomorphism can be chosen so that *f*_{p,q} is a rational
function. This is especially exciting when *K*_{p}
and *K*_{q} have
empty interiors, in that case γ_{p}
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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