Schwarz reflections and the Tricorn

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.

This is a sequel to the paper "Dynamics of Schwarz reflections: the mating phenomena", available at arXiv:1811.04979v3. Final version, to appear in "Ann. Inst. Fourier (Grenoble)"

References in corpus (4)

• Normal random matrix ensemble as a growth problem
• Dynamics of Schwarz reflections: the mating phenomena
• On a Family of Rational Perturbations of the Doubling Map
• Univalent Polynomials and Hubbard Trees

Cited by in corpus (5)

• On Dynamical Gaskets Generated by Rational Maps, Kleinian Groups, and Schwarz Reflections
• Univalent Polynomials and Hubbard Trees
• Schwarz reflections and anti-holomorphic correspondences
• Bers Slices in Families of Univalent Maps
• Uniformization of tongues in Double Standard Map family and variation of maximal chaotic sets