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تسجيل مستخدم جديدMating quadratic maps with the modular group III: The modular Mandelbrot set
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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 $mathcal{M}$.
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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 $Per_1(1)$.
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