Mating Siegel quadratic polynomials

Abstract

Let F F be a quadratic rational map of the sphere which has two fixed Siegel disks with bounded type rotation numbers θ \theta and ν \nu . Using a new degree 3 3 Blaschke product model for the dynamics of F F and an adaptation of complex a priori bounds for renormalization of critical circle maps, we prove that F F can be realized as the mating of two Siegel quadratic polynomials with the corresponding rotation numbers θ \theta and ν \nu .