## 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 - DOI: https://doi.org/10.1090/ecgd/379
- Published electronically: February 1, 2023
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## Abstract:

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.## References

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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: russell.lodge@indstate.edu
**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: mlyubich@math.stonybrook.edu
**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: smerenkov@ccny.cuny.edu
**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: sabya@math.tifr.res.in
- 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: https://doi.org/10.1090/ecgd/379
- MathSciNet review: 4567832