Convergence of pinching deformations and matings of geometrically finite polynomials

Peter Haïssinsky¹ ; Lei Tan²

¹ LATP//CMI Université de Provence 39, rue Frédéric Joliot-Curie 13453 Marseille Cedex 13, France
² Unité CNRS-UMR 8088 Département de Mathématiques Université de Cergy-Pontoise 2, avenue Adolphe Chauvin 95302 Cergy-Pontoise Cedex, France

Fundamenta Mathematicae, Tome 181 (2004) no. 2, pp. 143-188

Résumé

We give a thorough study of Cui's control of distortion technique in the analysis of convergence of simple pinching deformations, and extend his result from geometrically finite rational maps to some subset of geometrically infinite maps. We then combine this with mating techniques for pairs of polynomials to establish existence and continuity results for matings of polynomials with parabolic points. Consequently, if two hyperbolic quadratic polynomials tend to their respective root polynomials radially, and do not belong to conjugate limbs of the Mandelbrot set, then their mating exists and deforms continuously to the mating of the two root polynomials.

Détail

DOI : 10.4064/fm181-2-4

Keywords: thorough study cuis control distortion technique analysis convergence simple pinching deformations extend his result geometrically finite rational maps subset geometrically infinite maps combine mating techniques pairs polynomials establish existence continuity results matings polynomials parabolic points consequently hyperbolic quadratic polynomials tend their respective root polynomials radially belong conjugate limbs mandelbrot set their mating exists deforms continuously mating root polynomials