We prove that there exists a homeomorphism $chi$ between the connectedness locus $mathcal{M}_{Gamma}$ for the family $mathcal{F}_a$ of $(2:2)$ holomorphic correspondences introduced by Bullett and Penrose, and the parabolic Mandelbrot set $mathcal{M}
_1$. The homeomorphism $chi$ is dynamical ($mathcal{F}_a$ is a mating between $PSL(2,mathbb{Z})$ and $P_{chi(a)}$), it is conformal on the interior of $mathcal{M}_{Gamma}$, and it extends to a homeomorphism between suitably defined neighbourhoods in the respective one parameter moduli spaces. Following the recent proof by Petersen and Roesch that $mathcal{M}_1$ is homeomorphic to the classical Mandelbrot set $mathcal{M}$, we deduce that $mathcal{M}_{Gamma}$ is homeomorphic to $mathcal{M}$.
We show that the definition of parabolic-like map can be slightly modified, by asking $partial Delta$ to be a quasiarc out of the parabolic fixed point, instead of the dividing arcs to be $C^1$ on $[-1,0]$ and $[0,1]$.
We develop dynamical theory for the family of holomorphic correspondences $mathcal{F}_a$ proved by the current authors to be matings between the modular group and parabolic rational maps in the Milnor slice $Per_1(1)$ (in Mating quadratic maps with t
he modular group II). Such a mating endows the complement of the limit set of $mathcal{F}_a$ with the geometry of the hyperbolic plane, equipped with the action of the modular group. We introduce bi-infinite coding sequences for geodesics in this complement, utilising continued fraction expressions of end points; we prove landing theorems for periodic and preperiodic geodesics, and we establish a stronger Yoccoz inequality for repelling fixed points of these correspondences than Yoccozs classical inequality for quadratic polynomials. We deduce that the connectedness locus of the family $mathcal{F}_a$ is contained in a particular lune in parameter space.
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 $Per_1(1)$.
We prove that any $C^{1+BV}$ degree $d geq 2$ circle covering $h$ having all periodic orbits weakly expanding, is conjugate in the same smoothness class to a metrically expanding map. We use this to connect the space of parabolic external maps (comin
g from the theory of parabolic-like maps) to metrically expanding circle coverings.
