On Dynamical Gaskets Generated by Rational Maps, Kleinian Groups, and Schwarz Reflections

TL;DR

This paper explores the relationship between Kleinian groups, rational maps, and circle packings, revealing how certain limit sets and Julia sets are homeomorphic and sharing symmetry groups, with applications to classical gaskets and Schwarz reflections.

Contribution

It introduces a surgery connecting Kleinian groups generated by circle reflections to rational maps with homeomorphic Julia sets, and analyzes their symmetry groups and dynamical models.

Findings

Limit sets of Kleinian groups are homeomorphic to Julia sets of rational maps.

Symmetry groups of these sets are isomorphic and coincide with M"obius symmetry groups.

Constructs a mating between a group and a map using 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 M\"obius symmetries of $\Lambda_H$, which is the semi-direct product of $H$ itself and the group of M\"obius 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.