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

This is the authors' abstract. We don't add key points for this paper.

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 H whose limit set is a generalized Apollonian gasket Λ H Λ _H . We design a surgery that relates H H to a rational map g g whose Julia set J g J_g is (non-quasiconformally) homeomorphic to Λ H Λ _H . We show for a large class of triangulations, however, the groups of quasisymmetries of Λ H Λ _H and J g J_g are isomorphic and coincide with the corresponding groups of self-homeomorphisms. Moreover, in the case of H H , this group is equal to the group of Möbius symmetries of Λ H Λ _H , which is the semi-direct product of H H itself and the group of Möbius symmetries of the underlying circle packing. In the case of the tetrahedral triangulation (when Λ H Λ _H is the classical Apollonian gasket), we give a quasiregular model for the above actions which is quasiconformally equivalent to g g and produces H 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.