We give a new proof of the following conjecture of Yoccoz:
where , is its Siegel disk if is linearizable (or otherwise), is the conformal radius of the Siegel disk of (or if there is none) and is Yoccoz’s Brjuno function.
In a former article we obtained a first proof based on the control of parabolic explosion. Here, we present a more elementary proof based on Yoccoz’s initial methods.
We then extend this result to some new families of polynomials such as with . We also show that the conjecture does not hold for with .
Nous donnons une nouvelle preuve de la conjecture suivante de Yoccoz :
où , est son disque de Siegel si est linéarisable (ou sinon), est le rayon conforme du disque de Siegel de (ou s’il n’y en a pas) et est la fonction de Brjuno de Yoccoz.
Dans un article précédent nous avons obtenu une première preuve basée sur le contrôle de l’explosion parabolique. Ici, nous présentons une preuve plus élémentaire basée sur les méthodes initiales de Yoccoz.
Nous étendons ce résultat à quelques nouvelles familles de polynômes telle que avec . Nous montrons également que la conjecture ne tient pas pour avec .
Keywords: Siegel disks, quadratic polynomials, harmonic and subharbonic functions, conformal radius, holomorphic motions
Mot clés : disques de Siegel, polynômes quadratiques, fonctions harmoniques et sous-harmoniques, rayon conforme, mouvement holomorphe
@article{AIF_2011__61_1_319_0, author = {Buff, Xavier and Ch\'eritat, Arnaud}, title = {A new proof of a conjecture of {Yoccoz}}, journal = {Annales de l'Institut Fourier}, pages = {319--350}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {61}, number = {1}, year = {2011}, doi = {10.5802/aif.2603}, zbl = {1223.37061}, mrnumber = {2828132}, language = {en}, url = {http://www.numdam.org/articles/10.5802/aif.2603/} }
TY - JOUR AU - Buff, Xavier AU - Chéritat, Arnaud TI - A new proof of a conjecture of Yoccoz JO - Annales de l'Institut Fourier PY - 2011 SP - 319 EP - 350 VL - 61 IS - 1 PB - Association des Annales de l’institut Fourier UR - http://www.numdam.org/articles/10.5802/aif.2603/ DO - 10.5802/aif.2603 LA - en ID - AIF_2011__61_1_319_0 ER -
%0 Journal Article %A Buff, Xavier %A Chéritat, Arnaud %T A new proof of a conjecture of Yoccoz %J Annales de l'Institut Fourier %D 2011 %P 319-350 %V 61 %N 1 %I Association des Annales de l’institut Fourier %U http://www.numdam.org/articles/10.5802/aif.2603/ %R 10.5802/aif.2603 %G en %F AIF_2011__61_1_319_0
Buff, Xavier; Chéritat, Arnaud. A new proof of a conjecture of Yoccoz. Annales de l'Institut Fourier, Volume 61 (2011) no. 1, pp. 319-350. doi : 10.5802/aif.2603. http://www.numdam.org/articles/10.5802/aif.2603/
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