Kahn, Jeremy; Lyubich, Mikhail
A priori bounds for some infinitely renormalizable quadratics: II. Decorations  [ Bornes a priori pour quelques polynômes quadratiques infiniment renormalisables : II. Décorations ]
Annales scientifiques de l'École Normale Supérieure, Série 4 : Tome 41 (2008) no. 1 , p. 57-84
Zbl 1156.37311 | 1 citation dans Numdam
doi : 10.24033/asens.2063
URL stable : http://www.numdam.org/item?id=ASENS_2008_4_41_1_57_0

Une décoration de l’ensemble de Mandelbrot M est une partie de M découpée par deux rayons externes aboutissant à la pointe d’une petite copie de M 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.
A decoration of the Mandelbrot set M is a part of M cut off by two external rays landing at some tip of a satellite copy of M 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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