  We initiate the exploration of a new class of anti-holomorphic dynamical
systems generated by Schwarz reflection maps associated with quadrature
domains. More precisely, we study Schwarz reflection with respect to the
deltoid, and Schwarz reflections with respect to the cardioid and a family of
circumscribing circles. We describe the dynamical planes of the maps in
question, and show that in many cases, they arise as unique conformal matings
of quadratic anti-holomorphic polynomials and the ideal triangle group.
