Extremal univalent polynomials are classified by bi-angled trees and are in canonical bijection with anti-holomorphic polynomials with all critical points fixed.
Dynamics of Schwarz reflections: the mating phenomena
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abstract
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.
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Univalent Polynomials and Hubbard Trees
Extremal univalent polynomials are classified by bi-angled trees and are in canonical bijection with anti-holomorphic polynomials with all critical points fixed.