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 $\Psi (\mathcal{K})\subset \mathbb{H}$.

42 pages, 15 figures