Mating quadratic maps with the modular group II

TL;DR

This paper proves that all members of a specific family of holomorphic correspondences with connectedness locus are matings between the modular group and parabolic quadratic polynomials, extending previous conjectures.

Contribution

It establishes that every correspondence in the family with connectedness locus is a mating with the modular group and a parabolic quadratic polynomial, generalizing earlier results.

Findings

All members with connectedness locus are matings with the modular group.

Extension of Bullett and Penrose's conjecture to parabolic quadratic polynomials.

Confirmation that the family forms a rich class of dynamical systems.

Abstract

In 1994 S. Bullett and C. Penrose introduced the one complex parameter family of $(2:2)$ holomorphic correspondences $\mathcal{F}_a$: $$\left(\frac{aw-1}{w-1}\right)^2+\left(\frac{aw-1}{w-1}\right)\left(\frac{az+1}{z+1}\right) +\left(\frac{az+1}{z+1}\right)^2=3$$ and proved that for every value of $a \in [4,7] \subset \mathbb{R}$ the correspondence $\mathcal{F}_a$ is a mating between a quadratic polynomial $Q_c(z)=z^2+c,\,\,c \in \mathbb{R}$ and the modular group $\Gamma=PSL(2,\mathbb{Z})$. They conjectured that this is the case for every member of the family $\mathcal{F}_a$ which has $a$ in the connectedness locus. We prove here that every member of the family $\mathcal{F}_a$ which has $a$ in the connectedness locus is a mating between the modular group and an element of the parabolic quadratic family $Per_1(1)$.