ON DYNAMICAL GASKETS GENERATED BY RATIONAL MAPS, KLEINIAN GROUPS, AND SCHWARZ REFLECTIONS

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 ΛH. We design a surgery that relates H to a rational map g whose Julia set Jg is (non-quasiconformally) homeomorphic to ΛH. We show for a large class of triangulations, however, the groups of quasisymmetries of ΛH and Jg are isomorphic and coincide with the corresponding groups of selfhomeomorphisms. Moreover, in the case of H, this group is equal to the group of Möbius symmetries of ΛH, 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 ΛH 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.