On dynamical gaskets generated by rational maps, Kleinian groups, and Schwarz reflections

• Published in: Conformal geometry and dynamics 2023-02, Vol.27 (1), p.1-54
• Main Authors: Lodge, Russell, Lyubich, Mikhail, Merenkov, Sergei, Mukherjee, Sabyasachi
• Format: Article
• Language: English
• ISSN: 1088-4173; 1088-4173
• DOI: 10.1090/ecgd/379

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.