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 an Apollonian-like 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 piecewise affine 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.

54 pages, 14 figures, final accepted version

References in corpus (5)

• Dynamics of Schwarz reflections: the mating phenomena
• Circle packings, kissing reflection groups and critically fixed anti-rational maps
• Schwarz reflections and the Tricorn
• Schwarz reflections and anti-holomorphic correspondences
• David maps and Hausdorff Dimension

Cited by in corpus (4)

• Circle packings, kissing reflection groups and critically fixed anti-rational maps
• Antiholomorphic correspondences and mating I: realization theorems
• The Sullivan dictionary and Bowen-Series maps
• On quasiconformal non-equivalence of gasket Julia sets and limit sets