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
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.
@article{ASENS_2008_4_41_1_57_0,
     author = {Kahn, Jeremy and Lyubich, Mikhail Yu.},
     title = {A priori bounds for some infinitely renormalizable quadratics: II. Decorations},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     publisher = {Soci\'et\'e math\'ematique de France},
     volume = {Ser. 4, 41},
     number = {1},
     year = {2008},
     pages = {57-84},
     doi = {10.24033/asens.2063},
     zbl = {1156.37311},
     language = {en},
     url = {http://http://www.numdam.org/item/ASENS_2008_4_41_1_57_0}
}
Kahn, Jeremy; Lyubich, Mikhail. A priori bounds for some infinitely renormalizable quadratics: II. Decorations. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 41 (2008) no. 1, pp. 57-84. doi : 10.24033/asens.2063. http://www.numdam.org/item/ASENS_2008_4_41_1_57_0/

[1] L. V. Ahlfors, Conformal invariants: topics in geometric function theory, McGraw-Hill Book Co., 1973, McGraw-Hill Series in Higher Mathematics. | MR 357743 | Zbl 0272.30012

[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 1204.37047

[3] D. Cheraghi, Combinatorial rigidity for some infinitely renormalizable unicritical polynomials, preprint IMS at Stony Brook, #7, 2007. | MR 2719786 | Zbl 1218.37060

[4] A. Douady & J. H. Hubbard, On the dynamics of polynomial-like mappings, Ann. Sci. École Norm. Sup. 18 (1985), 287-343. | Numdam | Zbl 0587.30028

[5] F. P. Gardiner & N. Lakic, Quasiconformal Teichmüller theory, Mathematical Surveys and Monographs 76, Amer. Math. Soc., 2000. | Zbl 0949.30002

[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 1180.37072

[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 1203.30011

[9] M. Lyubich, Dynamics of quadratic polynomials. I, II, Acta Math. 178 (1997), 185-247, 247-297. | MR 1459261 | Zbl 0908.58053

[10] C. T. Mcmullen, Complex dynamics and renormalization, Annals of Mathematics Studies 135, Princeton University Press, 1994. | MR 1312365 | Zbl 0822.30002

[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 1765085 | Zbl 1107.37305

[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. | MR 1755445 | Zbl 0941.30016