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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-fm181-2-4,
author = {Peter Ha\"\i ssinsky and Lei Tan},
title = {Convergence of pinching deformations and matings of geometrically finite polynomials},
journal = {Fundamenta Mathematicae},
volume = {184},
year = {2004},
pages = {143-188},
zbl = {1048.37045},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm181-2-4}
}
Peter Haïssinsky; Lei Tan. Convergence of pinching deformations and matings of geometrically finite polynomials. Fundamenta Mathematicae, Tome 184 (2004) pp. 143-188. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm181-2-4/