Quadrature Domains and the Real Quadratic Family

TL;DR

This paper explores the connection between holomorphic dynamical systems, quadrature domains, and the Mandelbrot set, revealing how certain polynomial systems can be conformally mated with specific groups to form Schwarz functions of quadrature domains.

Contribution

It demonstrates that real-symmetric polynomials in the Mandelbrot set's hyperbolic component can be conformally mated with a subgroup of PSL(2,Z), forming Schwarz functions of quadrature domains.

Findings

Real-symmetric polynomials can be conformally mated with PSL(2,Z) subgroups.

The conformal mating corresponds to the Schwarz function of a quadrature domain.

Establishes a link between dynamical systems, group actions, and quadrature domains.

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.