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, Online first, 114 p.

Nous poursuivons l’exploration d’une famille 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 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 𝒮. Enfin, nous montrons que le modèle topologique locallement connexe du lieu de connexité de 𝒮 est naturellement homéomorphe à un tel modèle du membre associé à la basilique de la Tricorn.

We continue our exploration of the family 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 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 𝒮. Finally, we show that the locally connected topological model of the connectedness locus of 𝒮 is naturally homeomorphic to such a model of the basilica limb of the Tricorn.

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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) 
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Lee, Seung-Yeop; Lyubich, Mikhail; Makarov, Nikolai G.; Mukherjee, Sabyasachi. Schwarz reflections and the Tricorn. Annales de l'Institut Fourier, Online first, 114 p.

[1] Agam, O.; Bettelheim, E.; Wiegmann, P.; Zabrodin, A. Viscous Fingering and the Shape of an Electronic Droplet in the Quantum Hall Regime, Phys. Rev. Lett., Volume 88 (2002) no. 23, 236801 | DOI

[2] Aharonov, D.; Shapiro, H. S. A minimal-area problem in conformal mapping - preliminary report (1973) no. 7 (Research bulletin trita-mat)

[3] Aharonov, D.; Shapiro, H. S. Domains on which analytic functions satisfy quadrature identities, J. Anal. Math., Volume 30 (1976), pp. 39-73 | DOI | MR | Zbl

[4] Aharonov, D.; Shapiro, H. S. A minimal-area problem in conformal mapping - preliminary report. Part II (1978) no. 5 (Research bulletin trita-mat)

[5] Aharonov, D.; Shapiro, H. S.; Solynin, A. Y. A minimal area problem in conformal mapping, J. Anal. Math., Volume 78 (1999), pp. 157-176 | DOI | MR | Zbl

[6] Bergweiler, W.; Eremenko, A. Green’s function and anti-holomorphic dynamics on a torus, Proc. Am. Math. Soc., Volume 144 (2016) no. 7, pp. 2911-2922 | DOI | MR | Zbl

[7] Bogdanov, K.; Mamayusupov, K.; Mukherjee, S.; Schleicher, D. Antiholomorphic perturbations of Weierstrass Zeta functions and Green’s function on tori, Nonlinearity, Volume 30 (2017) no. 8, pp. 3241-3254 | DOI | MR | Zbl

[8] Bonifant, A.; Buff, X.; Milnor, J. Antipode preserving cubic maps: the fjord theorem, Proc. Lond. Math. Soc., Volume 116 (2018) no. 3, pp. 670-728 | DOI | MR | Zbl

[9] Branner, B.; Fagella, N. Quasiconformal surgery in holomorphic dynamics, Cambridge Studies in Advanced Mathematics, 141, Cambridge University Press, 2014 | DOI | MR | Zbl

[10] Bullett, S.; Lomonaco, L. Mating quadratic maps with the modular group II, Invent. Math., Volume 220 (2020) no. 1, pp. 185-210 | DOI | MR | Zbl

[11] Bullett, S.; Lomonaco, L. Dynamics of modular matings, Adv. Math., Volume 410 (2022) no. Part B, 108758 | DOI | MR | Zbl

[12] Bullett, S.; Penrose, C. Mating quadratic maps with the modular group, Invent. Math., Volume 115 (1994) no. 3, pp. 483-511 | DOI | MR | Zbl

[13] Caffarelli, L. A.; Karp, L.; Shahgholian, H. Regularity of a free boundary with application to the Pompeiu problem, Ann. Math., Volume 151 (2000) no. 1, pp. 269-292 | DOI | MR | Zbl

[14] Canela, J.; Fagella, N.; Garijo, A. On a family of rational perturbations of the doubling map, J. Difference Equ. Appl., Volume 21 (2015) no. 8, pp. 715-741 | DOI | MR | Zbl

[15] Carleson, L.; Gamelin, T. W. Complex Dynamics, Universitext, Springer, 1993 | DOI | MR | Zbl

[16] Crowe, W. D.; Hasson, R.; Rippon, P. J.; Clark, P. E. D. Strain On the Structure of the Mandelbar Set, Nonlinearity, Volume 2 (1989), pp. 541-553 | DOI | MR | Zbl

[17] Davis, P. J. The Schwarz function and its applications, The Carus Mathematical Monographs, 17, Mathematical Association of America, Buffalo, NY, 1974 | DOI | MR | Zbl

[18] Douady, A. Descriptions of compact sets in ℂ, Topological methods in modern mathematics (Stony Brook, NY, 1991), Publish or Perish, Inc., Houston, TX, 1993, pp. 429-465 | MR | Zbl

[19] Douady, A.; Hubbard, J. H. On the dynamics of polynomial-like mappings, Ann. Sci. Éc. Norm. Supér., Volume 18 (1985) no. 2, pp. 287-343 | DOI | Numdam | MR | Zbl

[20] Douady, Adrien; Hubbard, John H. Étude dynamique des polynômes complexes. Partie I, Publications Mathématiques d’Orsay, 84-2, Université de Paris-Sud, Département de Mathématiques, Orsay, 1984 | MR

[21] Douady, Adrien; Hubbard, John H. Étude dynamique des polynômes complexes. Partie II, Publications Mathématiques d’Orsay, 85-4, Université de Paris-Sud, Département de Mathématiques, Orsay, 1985 | MR

[22] Dudko, D. Matings with laminations (2011) | arXiv

[23] Duren, P. L. Univalent functions, Grundlehren der Mathematischen Wissenschaften, 259, Springer, 1983 | MR | Zbl

[24] Quadrature domains and their applications (Ebenfelt, P.; Gustafsson, B.; Khavinson, D.; Putinar, M., eds.), Operator Theory: Advances and Applications, 156, Birkhäuser, 2005 (The Harold S. Shapiro anniversary volume, Papers from the conference held at the University of California at Santa Barbara, Santa Barbara, CA, March 2003) | DOI | MR | Zbl

[25] Elbau, P.; Felder, G. Density of eigenvalues of random normal matrices, Commun. Math. Phys., Volume 259 (2005) no. 2, pp. 433-450 | DOI | MR | Zbl

[26] Gauthier, T.; Vigny, G. Distribution of postcritically finite polynomials III: Combinatorial continuity, Fundam. Math., Volume 244 (2019) no. 1, pp. 17-48 | DOI | MR | Zbl

[27] Goldberg, L. R. Fixed points of polynomial maps. I. Rotation subsets of the circles, Ann. Sci. Éc. Norm. Supér., Volume 25 (1992) no. 6, pp. 679-685 | DOI | Numdam | MR | Zbl

[28] Goldberg, L. R.; Milnor, J. Fixed points of polynomial maps II: Fixed point portraits, Ann. Sci. Éc. Norm. Supér., Volume 26 (1993), pp. 51-98 | DOI | Numdam | MR | Zbl

[29] Gustafsson, B. Quadrature identities and the Schottky double, Acta Appl. Math., Volume 1 (1983) no. 3, pp. 209-240 | DOI | MR | Zbl

[30] Gustafsson, B.; He, C.; Milanfar, P.; Putinar, M. Reconstructing planar domains from their moments, Inverse Probl., Volume 16 (2000) no. 4, pp. 1053-1070 | DOI | MR | Zbl

[31] Gustafsson, B.; Putinar, M. Hyponormal quantization of planar domains, Lecture Notes in Mathematics, 2199, Springer, 2017 (Exponential transform in dimension two) | DOI | MR | Zbl

[32] Gustafsson, B.; Vasilʼev, A. Conformal and potential analysis in Hele-Shaw cells, Advances in Mathematical Fluid Mechanics, Birkhäuser, 2006 | MR | Zbl

[33] Hatcher, A. Algebraic topology, Cambridge University Press, 2002 | MR | Zbl

[34] Hedenmalm, H.; Makarov, N. Coulomb gas ensembles and Laplacian growth, Proc. Lond. Math. Soc., Volume 106 (2013) no. 4, pp. 859-907 | DOI | MR | Zbl

[35] Hubbard, J. H. Local connectivity of Julia sets and bifurcation loci: three theorems of J.-C. Yoccoz, Topological methods in modern mathematics (Stony Brook, NY, 1991), Publish or Perish Inc., 1993, pp. 467-511 | MR | Zbl

[36] Hubbard, John; Schleicher, Dierk Multicorns are not Path Connected, Frontiers in Complex Dynamics. In Celebration of John Milnor’s 80th Birthday, Princeton University Press, 2014, pp. 73-102 | DOI | Zbl

[37] Inou, H. Combinatorics and topology of straightening maps II: Discontinuity. (2009) | arXiv

[38] Inou, H.; Mukherjee, S. Non-landing parameter rays of the multicorns, Invent. Math., Volume 204 (2016) no. 3, pp. 869-893 | DOI | MR | Zbl

[39] Inou, H.; Mukherjee, S. Discontinuity of straightening in anti-holomorphic dynamics: I, Trans. Am. Math. Soc., Volume 374 (2021) no. 9, pp. 6445-6481 | DOI | MR | Zbl

[40] Khavinson, D.; Świątek, G. On the number of zeros of certain harmonic polynomials, Proc. Am. Math. Soc., Volume 131 (2003) no. 2, pp. 409-414 | DOI | MR | Zbl

[41] Kiwi, J. Rational laminations of complex polynomials, Laminations and foliations in dynamics, geometry and topology (Stony Brook, NY, 1998) (Contemporary Mathematics), Volume 269, American Mathematical Society, 2001, pp. 111-154 | DOI | MR | Zbl

[42] Lazebnik, K.; Makarov, N. G.; Mukherjee, S. Univalent polynomials and Hubbard trees, Trans. Am. Math. Soc., Volume 374 (2021) no. 7, pp. 4839-4893 | DOI | MR | Zbl

[43] Lee, S.-Y.; Lyubich, M.; Makarov, N. G.; Mukherjee, S. Dynamics of Schwarz reflections: The mating phenomena, Ann. Sci. Ec. Norm. Supér. (4), Volume 56 (2023) no. 6, pp. 1825-1881 | MR | Zbl

[44] Lee, S.-Y.; Makarov, N. G. Topology of quadrature domains, J. Am. Math. Soc., Volume 29 (2016) no. 2, pp. 333-369 | DOI | MR | Zbl

[45] Lyubich, M. Dynamics of quadratic polynomials. I, II, Acta Math., Volume 178 (1997) no. 2, p. 185-247, 247–297 | DOI | MR | Zbl

[46] Lyubich, M. Conformal Geometry and Dynamics of Quadratic Polynomials, vol. I-II (2023) (http://www.math.stonybrook.edu/~mlyubich/book.pdf)

[47] Milnor, J. Remarks on iterated cubic maps, Exp. Math., Volume 1 (1992), pp. 5-24 | MR | Zbl

[48] Milnor, J. On rational maps with two critical points, Exp. Math., Volume 9 (2000) no. 4, pp. 481-522 | DOI | MR | Zbl

[49] Milnor, J. Periodic Orbits, External Rays and the Mandelbrot Set, Géométrie complexe et systèmes dynamiques (Astérisque), Volume 261, Société Mathématique de France, 2000, pp. 277-333 | Numdam | Zbl

[50] Milnor, J. Dynamics in one complex variable, Annals of Mathematics Studies, 160, Princeton University Press, 2006 | DOI | MR | Zbl

[51] Milnor, J. Hyperbolic components, Conformal dynamics and hyperbolic geometry (Contemporary Mathematics), Volume 573, American Mathematical Society, 2012, pp. 183-232 (With an appendix by A. Poirier) | DOI | MR | Zbl

[52] Moore, R. L. Concerning upper-semicontinuous collections of continua, Trans. Am. Math. Soc., Volume 27 (1925) no. 4, pp. 416-428 | DOI | MR | Zbl

[53] Mukherjee, S. Antiholomorphic Dynamics: Topology of Parameter Spaces, and Discontinuity of Straightening, Ph. D. Thesis, Jacobs University, Bremen, Germany (2015) (http://nbn-resolving.de/urn:nbn:de:gbv:579-opus-1005222)

[54] Mukherjee, S. Orbit portraits of unicritical antiholomorphic polynomials, Conform. Geom. Dyn., Volume 19 (2015), pp. 35-50 | DOI | MR | Zbl

[55] Mukherjee, S. Parabolic arcs of the multicorns: real-analyticity of Hausdorff dimension, and singularities of Per n (1) curves, Discrete Contin. Dyn. Syst., Volume 37 (2017) no. 5, pp. 2565-2588 | DOI | MR | Zbl

[56] Mukherjee, S.; Nakane, S.; Schleicher, D. On multicorns and unicorns II: bifurcations in spaces of antiholomorphic polynomials, Ergodic Theory Dyn. Syst., Volume 37 (2017) no. 3, pp. 859-899 | DOI | MR | Zbl

[57] Nakai, I. The classification of curvilinear angles in the complex plane and the groups of ± holomorphic diffeomorphisms, Ann. Fac. Sci. Toulouse, Math., Volume 7 (1998) no. 2, pp. 313-334 | DOI | Numdam | MR | Zbl

[58] Nakane, S. Connectedness of the Tricorn, Ergodic Theory Dyn. Syst., Volume 13 (1993), pp. 349-356 | DOI | MR | Zbl

[59] Nakane, S.; Schleicher, D. The nonarcwise-connectedness of the tricorn, Surikaisekikenkyusho Kokyuroku, Volume 959 (1996), pp. 73-83 Problems concerning complex dynamical systems (Japanese) (Kyoto, 1995) | MR

[60] Nakane, S.; Schleicher, D. On Multicorns and Unicorns I: Antiholomorphic Dynamics, Hyperbolic Components and Real Cubic Polynomials, Int. J. Bifurcation Chaos Appl. Sci. Eng., Volume 13 (2003), pp. 2825-2844 | DOI | MR | Zbl

[61] Poirier, A. Critical portraits for postcritically finite polynomials, Fundam. Math., Volume 203 (2009) no. 2, pp. 107-163 | DOI | MR | Zbl

[62] Richardson, S. Hele Shaw flows with a free boundary produced by the injection of fluid into a narrow channel, J. Fluid Mech., Volume 56 (1972) no. 4, pp. 609-618 | DOI | Zbl

[63] Saff, E. B.; Totik, V. Logarithmic potentials with external fields, Grundlehren der Mathematischen Wissenschaften, 316, Springer, 1997 (Appendix B by Thomas Bloom) | DOI | MR | Zbl

[64] Sakai, M. A moment problem on Jordan domains, Proc. Am. Math. Soc., Volume 70 (1978) no. 1, pp. 35-38 | DOI | MR | Zbl

[65] Sakai, M. Quadrature domains, Lecture Notes in Mathematics, 934, Springer, 1982 | DOI | MR | Zbl

[66] Sakai, M. Regularity of a boundary having a Schwarz function, Acta Math., Volume 166 (1991) no. 3-4, pp. 263-297 | DOI | MR | Zbl

[67] Schleicher, D. On fibers and local connectivity of Mandelbrot and Multibrot sets, Fractal geometry and applications: a jubilee of Benoît Mandelbrot. Part 1 (Proceedings of Symposia in Pure Mathematics), Volume 72, American Mathematical Society, 2004, pp. 477-517 | DOI | MR | Zbl

[68] Schleicher, Dierk Rational Parameter Rays of the Mandelbrot Set, Géométrie complexe et systèmes dynamiques (Astérisque), Volume 261, Société Mathématique de France, 2000, pp. 405-443 | Numdam | Zbl

[69] Shahgholian, H. On quadrature domains and the Schwarz potential, J. Math. Anal. Appl., Volume 171 (1992) no. 1, pp. 61-78 | DOI | MR | Zbl

[70] Sheil-Small, T. Complex polynomials, Cambridge Studies in Advanced Mathematics, 75, Cambridge University Press, 2002 | DOI | MR | Zbl

[71] Teodorescu, R.; Bettelheim, E.; Agam, O.; Zabrodin, A.; Wiegmann, P. Normal random matrix ensemble as a growth problem, Nucl. Phys., B, Volume 704 (2005) no. 3, pp. 407-444 | DOI | MR | Zbl

[72] Varchenko, A. N.; Ètingof, P. I. Why the boundary of a round drop becomes a curve of order four, University Lecture Series, 3, American Mathematical Society, 1992 | DOI | MR | Zbl

[73] Voronin, S. M. Analytic classification of pairs of involutions and its applications, Funkts. Anal. Prilozh., Volume 16 (1982) no. 2, pp. 21-29 | MR | Zbl

[74] Warschawski, S. E. On conformal mapping of infinite strips, Trans. Am. Math. Soc., Volume 51 (1942), pp. 280-335 | DOI | MR | Zbl

[75] Wiegmann, P. B. Aharonov–Bohm effect in the quantum Hall regime and Laplacian growth problems, Statistical field theories (Como, 2001) (NATO Science Series II: Mathematics, Physics and Chemistry), Volume 73, Kluwer Academic Publishers, 2002, pp. 337-349 | DOI | MR

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