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Schwarz reflections and anti-holomorphic correspondences

Lee, Seung-Yeop and Lyubich, Mikhail and Makarov, Nikolai G. and Mukherjee, Sabyasachi (2021) Schwarz reflections and anti-holomorphic correspondences. Advances in Mathematics, 385 . Art. No. 107766. ISSN 0001-8708. doi:10.1016/j.aim.2021.107766. https://resolver.caltech.edu/CaltechAUTHORS:20210504-074618137

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Abstract

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.


Item Type:Article
Related URLs:
URLURL TypeDescription
https://doi.org/10.1016/j.aim.2021.107766DOIArticle
https://arxiv.org/abs/1907.09107arXivDiscussion Paper
Additional Information:© 2021 Elsevier Inc. Received 10 October 2019, Revised 14 November 2020, Accepted 26 March 2021, Available online 22 April 2021.
Subject Keywords:Correspondences; Mating; Straightening map; Schwarz reflection
DOI:10.1016/j.aim.2021.107766
Record Number:CaltechAUTHORS:20210504-074618137
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20210504-074618137
Official Citation:Seung-Yeop Lee, Mikhail Lyubich, Nikolai G. Makarov, Sabyasachi Mukherjee, Schwarz reflections and anti-holomorphic correspondences, Advances in Mathematics, Volume 385, 2021, 107766, ISSN 0001-8708, https://doi.org/10.1016/j.aim.2021.107766.
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:108952
Collection:CaltechAUTHORS
Deposited By: Tony Diaz
Deposited On:05 May 2021 18:40
Last Modified:05 May 2021 18:40

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