Schwarz reflections and anti-holomorphic correspondences
Schwarz reflections and anti-holomorphic correspondences
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施瓦茨反射和反全纯对应
DOI：
10.1016/j.aim.2021.107766
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发表时间：
2019
影响因子：
1.7
通讯作者：
S. Mukherjee
中科院分区：
文献类型：
--
作者：
Seung;M. Lyubich;N. Makarov;S. Mukherjee
在本文中，我们继续探索的动力学和参数平面的单参数家庭的施瓦茨反射，开始在[14]，[15]。也就是说，我们考虑一个家庭的正交域限制切比雪夫三次多项式的各种单叶盘。然后，我们执行一个拟共形手术，将这些反射转化为抛物有理映射（这是我们理论的关键技术成分）。它在施瓦茨反射的参数平面和抛物Tricorn之间导出了一个拉直映射。我们描述了各种属性的拉直突出有关的问题，其反全纯性质。我们完成了讨论，通过比较我们的家庭与经典的Bullion-Penrose家庭的交配群和有理映射诱导的全纯对应。更确切地说，我们表明，施瓦茨反射引起的反全纯对应，是配对的抛物反有理映射的抽象模群。我们进一步说明我们的交配框架，通过研究与三角肌的施瓦茨反射图的对应关系。
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.
影响因子：
0.9
作者：
S. Mukherjee;S. Nakane;D. Schleicher
通讯作者：
S. Mukherjee;S. Nakane;D. Schleicher