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://archive.numdam.org/item/SB_2005-2006__48__385_0/}
}
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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://archive.numdam.org/item/SB_2005-2006__48__385_0/
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