CIRM - Videos & books Library - Interbreeding in conformal dynamics, and its applications near and far

Virtualconference

Auteurs : Mukherjee, Sabyasachi (Auteur de la conférence)
CIRM (Editeur )

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Résumé : Combination theorems play an important role in several areas of dynamics, geometry, and group theory. In this talk, we will expound a framework to conformally combine Kleinian (reflection) groups and (anti-)holomorphic rational maps in a single dynamical plane. In the anti-holomorphic setting, such hybrid dynamical systems are generated by Schwarz reflection maps arising from univalent rational maps. A crucial technical ingredient of this study is a recently developed David surgery technique that turns hyperbolic conformal dynamical systems to parabolic ones. We will also mention numerous consequences of this theory, including 1. an explicit dynamical connection between various rational Julia and Kleinian limit sets,2. existence of new classes of welding homeomorphisms and conformally removable Julia/limit sets, and3. failure of topological orbit equivalence rigidity for Fuchsian groups acting on the circle.

Mots-Clés : combination theorems; rational dynamics; reflection groups; Fuchsian groups; univalent maps; Schwarz reflection maps; univalent maps; Welding homeomorphisms; conformal removality; orbit equivalence

Codes MSC :
30C10 - Polynomials
30C45 - Special classes of univalent and multivalent functions (starlike, convex, bounded rotation, etc.)
30C50 - Coefficient problems for univalent and multivalent functions
30C62 - Quasiconformal mappings in the plane
30C75 - Extremal problems for conformal and quasiconformal mappings, other methods
30D05 - Functional equations in the complex domain, iteration and composition of analytic functions
30D40 - Cluster sets, prime ends, boundary behavior
30F40 - Kleinian groups, See also {20H10}
37F05 - Relations and correspondences
37F10 - Dynamics of complex polynomials, rational maps, entire and meromorphic functions; Fatou and Julia sets
37F20 - Combinatorics and topology in relation with holomorphic dynamical systems

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Hennenfent, Guillaume

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https://videos.cirm-math.fr/2021-09-24_Mukherjee.mp4

Informations sur la Rencontre

Nom de la Rencontre : Advancing Bridges in Complex Dynamics / Avancer les connections dans la dynamique complexe
Organisateurs de la Rencontre : Benini, Anna Miriam ; Drach, Kostiantyn ; Dudko, Dzmitry ; Hlushchanka, Mikhail ; Schleicher, Dierk
Dates : 20/09/2021 - 24/09/2021
Année de la rencontre : 2021
URL de la Rencontre : https://conferences.cirm-math.fr/2546.html

Données de citation

DOI : 10.24350/CIRM.V.19813803
Citer cette vidéo: Mukherjee, Sabyasachi (2021). Interbreeding in conformal dynamics, and its applications near and far. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.19813803
URI : http://dx.doi.org/10.24350/CIRM.V.19813803

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Bibliographie

S.-Y. Lee, M. Lyubich, N. G. Makarov, and S. Mukherjee, Dynamics of Schwarz reflections: the mating phenomena, https://arxiv.org/abs/1811.04979, 2018 - https://arxiv.org/abs/1811.04979

S.-Y. Lee, M. Lyubich, N. G. Makarov, and S. Mukherjee. Schwarz reflections and the Tricorn, https://arxiv.org/abs/1812.01573, 2018 - https://arxiv.org/abs/1812.01573

R. Lodge, M. Lyubich, S. Merenkov, and S. Mukherjee, On dynamical gaskets generated by rational maps, Kleinian groups, and Schwarz reflections, https://arxiv.org/abs/1912.13438, 2019. - https://arxiv.org/abs/1912.13438

R. Lodge, Y. Luo, and S. Mukherjee, Circle packings, kissing reflection groups, and critically fixed anti-rational maps, https://arxiv.org/abs/2007.03558, 2020. - https://arxiv.org/abs/2007.03558

M. Lyubich, S. Merenkov, D. Ntalampekos, and S. Mukherjee, David extension of circle homeomorphisms, welding, mating, and removability, https://arxiv.org/abs/2010.11256, 2020. - https://arxiv.org/abs/2010.1125

K. Lazebnik, N. G. Makarov, and S. Mukherjee, Univalent polynomials and Hubbard trees, Transactions of the American Mathematical Society, 374:4839-4893, 2021. - https://doi.org/10.1090/tran/8387

K. Lazebnik, N. G. Makarov, and S. Mukherjee, Bers slices in families of univalent maps, https://arxiv.org/abs/2007.02429, to appear in ‘Mathematische Zeitschrift', 2021. - https://arxiv.org/abs/2007.02429

S.-Y. Lee, M. Lyubich, N. G. Makarov, and S. Mukherjee, Schwarz reflections and anti-holomorphic correspondences, Advances in Mathematics, 385:107766, 2021. - https://doi.org/10.1016/j.aim.2021.107766

M. Mj and S. Mukherjee, Combination theorems in groups, geometry and dynamics, to appear in ‘In the tradition of Thurston II' (edited by Ohshika and Papadopoulos), 2021. - https://arxiv.org/abs/2103.10082

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