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Advancing Bridges in Complex Dynamics / Avancer les connections dans la dynamique complexe

Collection Advancing Bridges in Complex Dynamics / Avancer les connections dans la dynamique complexe

Organizer(s) Benini, Anna Miriam ; Drach, Kostiantyn ; Dudko, Dzmitry ; Hlushchanka, Mikhail ; Schleicher, Dierk
Date(s) 20/09/2021 - 24/09/2021
linked URL https://conferences.cirm-math.fr/2546.html
00:00:00 / 00:00:00
5 27

Interbreeding in conformal dynamics, and its applications near and far

By Sabyasachi Mukherjee

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.

Information about the video

Citation data

  • DOI 10.24350/CIRM.V.19813803
  • Cite this video Mukherjee, Sabyasachi (24/09/2021). Interbreeding in conformal dynamics, and its applications near and far. CIRM. Audiovisual resource. DOI: 10.24350/CIRM.V.19813803
  • URL https://dx.doi.org/10.24350/CIRM.V.19813803

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Bibliography

  • 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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