Schwarz reflections and the Tricorn

TL;DR

This paper explores the family of Schwarz reflection maps related to the cardioid and circle, establishing a deep connection with the basilica limb of the Tricorn through combinatorial and mating theories, and describing their topological models.

Contribution

It establishes a bijection between geometrically finite maps in the family and the basilica limb of the Tricorn, and describes their conformal matings and topological models.

Findings

A natural combinatorial bijection with the basilica limb of the Tricorn.

Every geometrically finite map arises as a conformal mating.

The topological model of the connectedness locus is homeomorphic to that of the basilica limb.

Abstract

We continue our exploration of the family $\mathcal{S}$ of Schwarz reflection maps with respect to the cardioid and a circle which was initiated in our earlier work. We prove that there is a natural combinatorial bijection between the geometrically finite maps of this family and those of the basilica limb of the Tricorn, which is the connectedness locus of quadratic anti-holomorphic polynomials. We also show that every geometrically finite map in $\mathcal{S}$ arises as a conformal mating of a unique geometrically finite quadratic anti-holomorphic polynomial and a reflection map arising from the ideal triangle group. We then follow up with a combinatorial mating description for the periodically repelling maps in $\mathcal{S}$. Finally, we show that the locally connected topological model of the connectedness locus of $\mathcal{S}$ is naturally homeomorphic to such a model of the basilica limb of the Tricorn.