A priori bounds for some infinitely renormalizable quadratics: II. Decorations

Kahn, Jeremy; Lyubich, Mikhail

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 41 (2008) no. 1, pp. 57-84.

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.

[1] L. V. Ahlfors, Conformal invariants: topics in geometric function theory, McGraw-Hill Book Co., 1973, McGraw-Hill Series in Higher Mathematics. | MR | Zbl

[2] A. Avila, J. Kahn, M. Lyubich & W. Shen, Combinatorial rigidity for unicritical polynomials, preprint IMS at Stony Brook, #5, 2005. To appear in Annals of Math.. | Zbl

[3] D. Cheraghi, Combinatorial rigidity for some infinitely renormalizable unicritical polynomials, preprint IMS at Stony Brook, #7, 2007. | MR | Zbl

[4] A. Douady & J. H. Hubbard, On the dynamics of polynomial-like mappings, Ann. Sci. École Norm. Sup. 18 (1985), 287-343. | EuDML | Numdam | Zbl

[5] F. P. Gardiner & N. Lakic, Quasiconformal Teichmüller theory, Mathematical Surveys and Monographs 76, Amer. Math. Soc., 2000. | Zbl

[6] J. Kahn, A priori bounds for some infinitely renormalizable quadratics: I. bounded primitive combinatorics, preprint IMS at Stony Brook, #5, 2006.

[7] J. Kahn & M. Lyubich, Local connectivity of Julia sets for unicritical polynomials, preprint IMS at Stony Brook, #3, 2005. To appear in Annals of Math.. | Zbl

[8] J. Kahn & M. Lyubich, Quasi-additivity law in conformal geometry, preprint IMS at Stony Brook, #2, arXiv:math.DS/0505191v2, 2005. To appear in Annals of Math.. | Zbl

[9] M. Lyubich, Dynamics of quadratic polynomials. I, II, Acta Math. 178 (1997), 185-247, 247-297. | MR | Zbl

[10] C. T. Mcmullen, Complex dynamics and renormalization, Annals of Mathematics Studies 135, Princeton University Press, 1994. | MR | Zbl

[11] J. Milnor, Local connectivity of Julia sets: expository lectures, in The Mandelbrot set, theme and variations, London Math. Soc. Lecture Note Ser. 274, Cambridge Univ. Press, 2000, 67-116. | MR | Zbl

[12] J. Milnor, Periodic orbits, externals rays and the Mandelbrot set: an expository account (Géométrie complexe et systèmes dynamiques, Orsay, 1995), Astérisque 261 (2000), 277-333. | Numdam | MR | Zbl