Xavier Buff et Arnaud Chéritat ont montré que l'ensemble de Julia de certains polynômes quadratiques est de mesure de Lebesgue positive, répondant ainsi à une question ouverte depuis Fatou et Julia. Les polynômes en question ont un point fixe indifférent irrationnel dont le nombre de rotation doit être soigneusement déterminé. On exposera les grandes lignes de la démonstration, ainsi que d'autres résultats connexes des mêmes auteurs sur la géométrie et la taille des disques de Siegel.
Xavier Buff and Arnaud Chéritat have shown that the Julia sets of some quadratic polynomials have positive Lebesgue measure, answering a question open since Fatou and Julia. These polynomials have an indifferent fixed point with carefully selected rotation number. We will explain the main steps of their proof and present related results of the same authors on the geometry and the size of Siegel disks.
Mot clés : ensembles de Julia, disques de Siegel, dynamique holomorphe
Keywords: Julia sets, Siegel disks, holomorphic dynamics
@incollection{SB_2005-2006__48__385_0,
author = {Yoccoz, Jean-Christophe},
title = {Ensembles de {Julia} de mesure positive et disques de {Siegel} des polyn\^omes quadratiques},
booktitle = {S\'eminaire Bourbaki : volume 2005/2006, expos\'es 952-966},
series = {Ast\'erisque},
note = {talk:966},
pages = {385--401},
publisher = {Soci\'et\'e math\'ematique de France},
number = {311},
year = {2007},
mrnumber = {2359051},
zbl = {1194.37072},
language = {fr},
url = {http://www.numdam.org/item/SB_2005-2006__48__385_0/}
}
TY - CHAP AU - Yoccoz, Jean-Christophe TI - Ensembles de Julia de mesure positive et disques de Siegel des polynômes quadratiques BT - Séminaire Bourbaki : volume 2005/2006, exposés 952-966 AU - Collectif T3 - Astérisque N1 - talk:966 PY - 2007 SP - 385 EP - 401 IS - 311 PB - Société mathématique de France UR - http://www.numdam.org/item/SB_2005-2006__48__385_0/ LA - fr ID - SB_2005-2006__48__385_0 ER -
%0 Book Section %A Yoccoz, Jean-Christophe %T Ensembles de Julia de mesure positive et disques de Siegel des polynômes quadratiques %B Séminaire Bourbaki : volume 2005/2006, exposés 952-966 %A Collectif %S Astérisque %Z talk:966 %D 2007 %P 385-401 %N 311 %I Société mathématique de France %U http://www.numdam.org/item/SB_2005-2006__48__385_0/ %G fr %F SB_2005-2006__48__385_0
Yoccoz, Jean-Christophe. Ensembles de Julia de mesure positive et disques de Siegel des polynômes quadratiques, dans Séminaire Bourbaki : volume 2005/2006, exposés 952-966, Astérisque, no. 311 (2007), Exposé no. 966, pp. 385-401. http://www.numdam.org/item/SB_2005-2006__48__385_0/
[1] , & - “Siegel disks with smooth boundaries”, Acta Math. 193 (2004), no. 1, p. 1-30. | MR | Zbl
[2] - “Analytic form of differential equations I, II”, Trudy Moskov. Mat. Obšč. 25, 26 (1971, 1972), p. 119-262, p. 199-239. | MR | Zbl
[3] - “Disques de Siegel et ensembles de Julia d'aire strictement positive”, preprint, http://www.picard.ups-tlse.fr/~buff/HDR/HDR.pdf.
[4] & - “The Brjuno function continuously estimates the size of quadratic Siegel disks”, Ann. of Math. (2) 164 (2006), p. 265-312. | MR | Zbl
[5] -, “Quadratic Julia sets with positive area”, preprint http://arxiv.org/abs/math/0605514. | Zbl
[6] -, “Upper bound for the size of quadratic Siegel disks”, Invent. Math. 156 (2004), no. 1, p. 1-24. | MR | Zbl
[7] -, “Ensembles de Julia quadratiques de mesure de Lebesgue strictement positive”, C. R. Math. Acad. Sci. Paris 341 (2005), no. 11, p. 669-674. | MR | Zbl
[8] -, “How regular can the boundary of a quadratic Siegel disk be ?”, Proc. Amer. Math. Soc. 135 (2007), p. 1073-1080. | DOI | MR | Zbl
[9] - “Recherche d'ensembles de Julia de mesure de Lebesgue positive”, Thèse, Orsay, 2001.
[10] -, “Sur la vitesse d'explosion des points fixes paraboliques dans la famille quadratique”, C. R. Math. Acad. Sci. Paris 334 (2002), no. 12, p. 1107-1112. | MR | Zbl
[11] & - “Étude dynamique des polynômes complexes I, II”, Publ. Math. Orsay (1984-85). | Zbl
[12] - “Smooth Siegel discs without number theory : A remark on a proof by Buff and Chéritat”, preprint http://arxiv.org/abs/math.DS/0510578 (2003). | MR | Zbl
[13] & - “The renormalization for parabolic fixed points and their perturbation”, preprint, http://www.math.kyoto-u.ac.jp/~mitsu/pararenorm/ParabolicRenormalization.pdf.
[14] - “Perturbation d'une fonction linéarisable”, in The Mandelbrot set, theme and variations, London Math. Soc. Lecture Note Ser., vol. 274, Cambridge Univ. Press, Cambridge, 2000, p. 227-252. | MR | Zbl
[15] - “On the Lebesgue measure of the Julia set of a quadratic polynomial”, Stonybrook IMS, preprint (1991/10).
[16] - “Critical functions for complex analytic maps”, J. Phys. A 23 (1990), no. 15, p. 3447-3474. | MR | Zbl
[17] , & - “The Brjuno functions and their regularity properties”, Comm. Math. Phys. 186 (1997), no. 2, p. 265-293. | MR | Zbl
[18] - “Self-similarity of Siegel disks and Hausdorff dimension of Julia sets”, Acta Math. 180 (1998), no. 2, p. 247-292. | MR | Zbl
[19] - “Siegel disks with smooth boundary”, preprint (1997).
[20] & - “On the Julia set of a typical quadratic polynomial with a Siegel disk”, Ann. of Math. (2) 159 (2004), no. 1, p. 1-52. | MR | Zbl
[21] - Unpublished.
[22] - “Iteration of analytic functions”, Ann. of Math. (2) 43 (1942), p. 607-612. | MR | Zbl
[23] - Petits diviseurs en dimension , Astérisque, vol. 231, Soc. Math. France, Paris, 1995. | Numdam | Zbl
[24] -, “Analytic linearization of circle diffeomorphisms”, in Dynamical systems and small divisors (Cetraro, 1998), Lecture Notes in Math., vol. 1784, Springer, Berlin, 2002, p. 125-173. | MR | Zbl
