A domain in the complex plane is called a quadrature domain if it admits a global Schwarz reflection map. Topology of quadrature domains has important applications to physics, and is intimately related to iteration of Schwarz reflection maps. As dynamical systems, Schwarz reflection maps produce various instances of “matings” of rational maps and groups.

We will introduce this new class of dynamical systems, and illustrate the above-mentioned mating phenomenon with a few concrete examples. We will then describe a specific one-parameter family of Schwarz reflection maps such that “typical” maps in this family arise as unique conformal matings of a quadratic anti-holomorphic polynomial and the ideal triangle group.

Time permitting, we will also mention how Schwarz reflection maps provide us with a framework for constructing correspondences on the Riemann sphere that are “matings” of rational maps and groups.

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Last updated: 23 Jun 2024