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

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.