no. 5
Schwarz reflections and the Tricorn
[Sur les réflections de Schwarz et la Tricorn]
Lee, Seung-Yeop1 ; Lyubich, Mikhail2 ; Makarov, Nikolai G.3 ; Mukherjee, Sabyasachi4
1 Department of Mathematics and Statistics, University of South Florida, Tampa, FL 33620 (USA)
2 Institute for Mathematical Sciences, Stony Brook University, NY, 11794 (USA)
3 Department of Mathematics, California Institute of Technology, Pasadena, California 91125 (USA)
4 School of Mathematics, Tata Institute of Fundamental Research, 1 Homi Bhabha Road, Mumbai 400005 (India)
Annales de l'Institut Fourier, Tome 75 (2025) no. 5, pp. 1987-2100

We continue our exploration of the family $\mathcal{S}$ of Schwarz reflection maps with respect to the cardioid and a circle which was initiated in our earlier work. We prove that there is a natural combinatorial bijection between the geometrically finite maps of this family and those of the basilica limb of the Tricorn, which is the connectedness locus of quadratic anti-holomorphic polynomials. We also show that every geometrically finite map in $\mathcal{S}$ arises as a conformal mating of a unique geometrically finite quadratic anti-holomorphic polynomial and a reflection map arising from the ideal triangle group. We then follow up with a combinatorial mating description for the periodically repelling maps in $\mathcal{S}$. Finally, we show that the locally connected topological model of the connectedness locus of $\mathcal{S}$ is naturally homeomorphic to such a model of the basilica limb of the Tricorn.

Nous poursuivons l’exploration d’une famille $\mathcal{S}$ d’applications de réflections de Schwarz par rapport la cardioïde et par rapport au cercle, qui a été initiée dans des travaux antérieurs. Nous prouvons qu’il y a une bijection naturelle de nature combinatoire entre les applications géométriquement finies de cette famille et celles du membre associé à la basilique de la Tricorn, qui est le lieu de connexité des applications polynomiales anti-holomorphes de degré deux. Nous prouvons aussi que toute application géométriquement finie dans $\mathcal{S}$ provient d’un accouplement conforme entre un polynôme quadratique anti-holomorphe géométriquement fini uniquement déterminé avec une application associée à un groupe engendré par les réflections par rapport aux côtés d’un triangle idéal. Nous continuons avec une description d’un accouplement combinatoire pour les applications périodiquement répulsives de $\mathcal{S}$. Enfin, nous montrons que le modèle topologique locallement connexe du lieu de connexité de $\mathcal{S}$ est naturellement homéomorphe à un tel modèle du membre associé à la basilique de la Tricorn.

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Révisé le :
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DOI : 10.5802/aif.3700
Classification : 37F10, 37F20, 37F31, 37F32, 37F46, 30C45, 30D05, 30D40, 82B21
Keywords: Schwarz reflection map, Antiholomorphic dynamics, Mating, Reflection group, Quadrature domain
Mots-clés : Réflections de Schwarz, Dynamique d’application anti-holomorphes, Accouplement, Groupe de réflection, Domaine de quadrature

Lee, Seung-Yeop 1 ; Lyubich, Mikhail 2 ; Makarov, Nikolai G. 3 ; Mukherjee, Sabyasachi 4

1 Department of Mathematics and Statistics, University of South Florida, Tampa, FL 33620 (USA)
2 Institute for Mathematical Sciences, Stony Brook University, NY, 11794 (USA)
3 Department of Mathematics, California Institute of Technology, Pasadena, California 91125 (USA)
4 School of Mathematics, Tata Institute of Fundamental Research, 1 Homi Bhabha Road, Mumbai 400005 (India)
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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Lee, Seung-Yeop; Lyubich, Mikhail; Makarov, Nikolai G.; Mukherjee, Sabyasachi. Schwarz reflections and the Tricorn. Annales de l'Institut Fourier, Tome 75 (2025) no. 5, pp. 1987-2100. doi: 10.5802/aif.3700

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