In this paper, we continue exploration of the dynamical and parameter planes of one-parameter families of Schwarz reflections that was initiated in [14], [15]. Namely, we consider a family of quadrature domains obtained by restricting the Chebyshev cubic polynomial to various univalent discs. Then we perform a quasiconformal surgery that turns these reflections to parabolic rational maps (which is the crucial technical ingredient of our theory). It induces a straightening map between the parameter plane of Schwarz reflections and the parabolic Tricorn. We describe various properties of this straightening highlighting the issues related to its anti-holomorphic nature. We complete the discussion by comparing our family with the classical Bullett-Penrose family of matings between groups and rational maps induced by holomorphic correspondences. More precisely, we show that the Schwarz reflections give rise to anti-holomorphic correspondences that are matings of parabolic anti-rational maps with the abstract modular group. We further illustrate our mating framework by studying the correspondence associated with the Schwarz reflection map of a deltoid.

在本文中，我们继续探索始于[14]，[15]的Schwarz反射一参数族的动力学和参数平面。即，我们考虑通过将切比雪夫三次多项式限制为各种单价光盘而获得的一族正交域。然后，我们执行准共形外科手术，将这些反射转变为抛物线有理图（这是我们理论的关键技术要素）。它在Schwarz反射的参数平面和抛物线Tricorn之间产生一个拉直映射。我们描述了这种矫直的各种特性，突出了与其反全同性有关的问题。我们通过将我们的家庭与经典的Bullett-Penrose家庭在群体之间的交配和由全纯对应引起的有理图进行比较来完成讨论。更准确地说，我们证明了Schwarz反射产生了反全同的对应关系，该对应关系是抛物线的反理性映射与抽象的模块化组的匹配。通过研究与三角肌的Schwarz反射图相关的对应关系，我们进一步说明了我们的交配框架。