Tessellating the discreteness locus for the modular mating family of correspondences

Abstract

The modular Mandelbrot set $M_\Gamma$, the connectedness locus of the modular mating family of 2 : 2 holomorphic correspondences $\mathcal{F}_a$ on the Riemann sphere, is homeomorphic to the classical Mandelbrot set $M$. The Klein combination locus $\mathcal{K}$ (the "discreteness locus" of the family $\mathcal{F}_a$) is a pinched neighborhood of $M_\Gamma$ in the $a$-plane, pinched at the root point. We construct a canonical map $\Psi$ from $\mathcal{K}\setminus M_\Gamma$ into the hyperbolic plane $\mathbb{H}$, inspired by the construction of Douady and Hubbard for their celebrated conformal bijection $\Phi: \mathbb{C}\setminus M \to \mathbb{C}\setminus \overline{\mathbb{D}}$, and we prove that $\Psi$ is analytic. This map $\Psi$ induces a tessellation of $\mathcal{K}\setminus M_\Gamma$ by pulling back a tessellation of $\mathbb{H}$ invariant under the modular group. We develop a series of conjectures concerning the structure of $\mathcal{K}$, its boundary, and $\Psi(\mathcal{K}) \subset \mathbb{H}$.