The Parabolic Mandelbrot Set

We solve the longstanding conjecture by Milnor (1993) concerning the connectedness locus \(M_1\) of the family of quadratic rational maps tangent to the identity at \(\infty\). We prove that this locus in homeomorphic to the Mandelbrot set \(M\) and that the homeomorphism is unique, provided it identifies maps that are "hybridly" conjugate on their filled-in Julia set. Moreover this homeomorphism from \(M\) to \(M_1\) is nowhere H\"older on the boundary and so can not have even locally a quasi-conformal extension to complements.