Quadrature Domains and the Real Quadratic Family

Abstract

We study several classes of holomorphic dynamical systems associated with quadrature domains. Our main result is that real-symmetric polynomials in the principal hyperbolic component of the Mandelbrot set can be conformally mated with a congruence subgroup of $\textrm{PSL}(2,\mathbb{Z})$, and that this conformal mating is the Schwarz function of a simply connected quadrature domain.