Mating quadratic maps with the modular group III: The modular Mandelbrot set

TL;DR

This paper establishes a homeomorphism between a family of holomorphic correspondences related to quadratic maps and the classical Mandelbrot set, revealing deep structural similarities and extending known results in complex dynamics.

Contribution

It proves the existence of a dynamical and conformal homeomorphism between the connectedness locus of certain holomorphic correspondences and the parabolic Mandelbrot set, linking these complex structures.

Findings

Homeomorphism between $ ext{M}_ ext{Gamma}$ and $ ext{M}_1$

Extension of homeomorphism to moduli spaces

$ ext{M}_ ext{Gamma}$ is homeomorphic to the classical Mandelbrot set

Abstract

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 $\mathcal{M}$.