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Conformal Geometry and Dynamics

Published by the American Mathematical Society since 1997, the purpose of this electronic-only journal is to provide a forum for mathematical work in related fields broadly described as conformal geometry and dynamics. All articles are freely available to all readers and with no publishing fees for authors.

ISSN 1088-4173

The 2020 MCQ for Conformal Geometry and Dynamics is 0.49.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.


On dynamical gaskets generated by rational maps, Kleinian groups, and Schwarz reflections
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by Russell Lodge, Mikhail Lyubich, Sergei Merenkov and Sabyasachi Mukherjee
Conform. Geom. Dyn. 27 (2023), 1-54
Published electronically: February 1, 2023


According to the Circle Packing Theorem, any triangulation of the Riemann sphere can be realized as a nerve of a circle packing. Reflections in the dual circles generate a Kleinian group $H$ whose limit set is a generalized Apollonian gasket $\Lambda _H$. We design a surgery that relates $H$ to a rational map $g$ whose Julia set $\mathcal {J}_g$ is (non-quasiconformally) homeomorphic to $\Lambda _H$. We show for a large class of triangulations, however, the groups of quasisymmetries of $\Lambda _H$ and $\mathcal {J}_g$ are isomorphic and coincide with the corresponding groups of self-homeomorphisms. Moreover, in the case of $H$, this group is equal to the group of Möbius symmetries of $\Lambda _H$, which is the semi-direct product of $H$ itself and the group of Möbius symmetries of the underlying circle packing. In the case of the tetrahedral triangulation (when $\Lambda _H$ is the classical Apollonian gasket), we give a quasiregular model for the above actions which is quasiconformally equivalent to $g$ and produces $H$ by a David surgery. We also construct a mating between the group and the map coexisting in the same dynamical plane and show that it can be generated by Schwarz reflections in the deltoid and the inscribed circle.
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Bibliographic Information
  • Russell Lodge
  • Affiliation: Department of Mathematics and Computer Science, Indiana State University, Terre Haute, Indiana 47809
  • MR Author ID: 1022713
  • Email:
  • Mikhail Lyubich
  • Affiliation: Mathematics Department and the Institute for Mathematical Sciences, Stony Brook University, 100 Nicolls Road, Stony Brook, New York 11794
  • MR Author ID: 189401
  • Email:
  • Sergei Merenkov
  • Affiliation: Department of Mathematics, City College of New York, New York, New York 10031; and Mathematics Program, CUNY Graduate Center, New York, New York 10016.
  • MR Author ID: 663502
  • ORCID: 0000-0002-6336-4640
  • Email:
  • Sabyasachi Mukherjee
  • Affiliation: School of Mathematics, Tata Institute of Fundamental Research, 1 Homi Bhabha Road, Mumbai 400005, India
  • MR Author ID: 1098266
  • ORCID: 0000-0002-6868-6761
  • Email:
  • Received by editor(s): November 8, 2021
  • Received by editor(s) in revised form: September 26, 2022
  • Published electronically: February 1, 2023
  • Additional Notes: The second author was supported by NSF grants DMS-1600519 and 1901357, and by a Fellowship from the Hagler Institute for Advanced Study. The third author was supported by NSF grant DMS-1800180. The fourth author was supported by the Department of Atomic Energy, Government of India, under project no. 12-R&D-TFR-5.01-0500, an endowment of the Infosys Foundation, and SERB research project grant SRG/2020/000018. The first and last authors were supported by the Institute for Mathematical Sciences at Stony Brook University during part of the work on this project.
  • © Copyright 2023 American Mathematical Society
  • Journal: Conform. Geom. Dyn. 27 (2023), 1-54
  • MSC (2020): Primary 37F10, 37F32, 30C62, 30C10, 30D05; Secondary 30F40, 30C45, 37F31
  • DOI:
  • MathSciNet review: 4567832