A priori bounds for some infinitely renormalizable quadratics: II. Decorations
Jeremy KAHN, Mikhail LYUBICH
ANNALES SCIENTIFIQUES DE L’ÉCOLE NORMALE SUPÉRIEURE, série 4, 41,
fascicule 1 (2008), 57-84
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Résumé :
Bornes a priori pour quelques polynômes quadratiques infiniment
renormalisables : II. Décorations
Une décoration de l’ensemble de Mandelbrot 
est une partie de 
découpée par
deux rayons externes aboutissant à la pointe d’une petite copie de 
attachée à la
cardioïde principale. Dans cet article nous considérons des polynômes quadratiques
infiniment renormalisables qui satisfont à la condition de décoration, à savoir que la
combinatoire des opérateurs de renormalisation mis en jeu est sélectionnée à partir
d’une famille finie de décorations. Pour cette classe d’applications, nous donnons des
bornes a priori. Ces bornes impliquent la connexité locale des ensembles de
Julia correspondants et celle de l’ensemble de Mandelbrot aux paramètres
correspondants.
Abstract :
A decoration of the Mandelbrot set 
is a part of 
cut off by two external rays
landing at some tip of a satellite copy of 
attached to the main cardioid. In this
paper we consider infinitely renormalizable quadratic polynomials satisfying the
decoration condition, which means that the combinatorics of the renormalization
operators involved is selected from a finite family of decorations. For this class
of maps we prove a priori bounds. They imply local connectivity of the
corresponding Julia sets and the Mandelbrot set at the corresponding parameter
values.
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