Dynamics of Schwarz reflections: the mating phenomena
Dynamical Systems
2025-05-08 v3 Complex Variables
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.
Cite
@article{arxiv.1811.04979,
title = {Dynamics of Schwarz reflections: the mating phenomena},
author = {Seung-Yeop Lee and Mikhail Lyubich and Nikolai G. Makarov and Sabyasachi Mukherjee},
journal= {arXiv preprint arXiv:1811.04979},
year = {2025}
}
Comments
Same as accepted version, to appear in "Ann. Sci. \'Ec. Norm. Sup\'er. (4)"