Mating Siegel Quadratic Polynomials

1.1. Mating: Definitions and some history. Mating quadratic polynomials is a topological construction suggested by Douady and Hubbard [Do2] to partially parametrize quadratic rational maps of the Riemann sphere by pairs of quadratic polynomials. Some results on matings of higher degree maps exist, but we will not discuss them in this paper. While there exist several, presumably equivalent, ways of describing the construction of mating, the following approach is perhaps the most standard. Consider two monic quadratic polynomials f1 and f2 whose filled Julia sets K(fi) are locally-connected. For each fi, let Φi denote the conformal isomorphism between the basin of infinity Ĉ rK(fi) and Ĉ r D, with Φi(∞) =∞ and Φi(∞) = 1. These Böttcher maps conjugate the polynomials to the squaring map: