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Abstract
In 1994 S. Bullett and C. Penrose introduced the one complex parameter family of (2 : 2) holomorphic correspondences mathcal {F}_a: aw-1w-12+aw-1w-1az+1z+1+az+1z+12=3\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$\\begin{aligned} \\left( \\frac{aw-1}{w-1}\\right) ^2+\\left( \\frac{aw-1}{w-1}\\right) \\left( \\frac{az+1}{z+1}\\right) +\\left( \\frac{az+1}{z+1}\\right) ^2=3 \\end{aligned}$$\\end{document}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 varGamma =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 show here that matings between the modular group and rational maps in the parabolic quadratic family Per_1(1) provide a better model: we prove that every member of the family mathcal {F}_a which has a in the connectedness locus is such a mating.
Highlights
We show here that matings between the modular group and rational maps in the parabolic quadratic family Per1(1) provide a better model: we prove that every member of the family Fa which has a in the connectedness locus is such a mating
The study of iterated holomorphic correspondences was initiated by Fatou [11] in 1922, with an analysis of a family of examples ‘sur lesquels’, he remarks, ‘on voit apparaitre déjà certaines propriétés, assez différentes de celles auxquelles donnent lieu les cas d’itération déjà étudiés’ [‘in which one already sees the appearance of certain properties somewhat different from those arising in the cases of iteration studied up till now’]
The developments of which we are aware came in the 1990s, when McMullen and Sullivan in their foundational work [17], defined a holomorphic dynamical system to be a collection of holomorphic relations on a complex 1-manifold, and developed a common framework in which rational maps, Kleinian groups and holomorphic correspondences can be treated simultaneously
Summary
Proposition 2.1 For every holomorphic function g as above, and every angle 0 < θ < π , inside every neighbourhood of 0 there exists a repelling petal Uθ+ containing an open sector of angle 2θ centered at the origin and symmetric with respect to the repelling direction. Each of these petals is equipped with a conformal subset Vθ+. Theorem for parabolic-like maps, any degree 2 parabolic-like map is hybrid equivalent to a member of the family Per1(1), a unique such member if the filled Julia set is connected
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