# ANNALES DE L'INSTITUT FOURIER

A new proof of a conjecture of Yoccoz
Annales de l'Institut Fourier, Volume 61 (2011) no. 1, p. 319-350
We give a new proof of the following conjecture of Yoccoz:$\left(\exists C\in ℝ\right)\phantom{\rule{3.33333pt}{0ex}}\left(\forall \theta \in ℝ\setminus ℚ\right)\phantom{\rule{1em}{0ex}}log\mathrm{rad}\phantom{\rule{0.166667em}{0ex}}\Delta \left({Q}_{\theta }\right)\le -Y\left(\theta \right)+C,$where ${Q}_{\theta }\left(z\right)={\mathrm{e}}^{2\pi i\theta }z+{z}^{2}$, $\Delta \left({Q}_{\theta }\right)$ is its Siegel disk if ${Q}_{\theta }$ is linearizable (or $\varnothing$ otherwise), $\mathrm{rad}\phantom{\rule{0.166667em}{0ex}}\Delta \left({Q}_{\theta }\right)$ is the conformal radius of the Siegel disk of ${Q}_{\theta }$ (or $0$ if there is none) and $Y\left(\theta \right)$ 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 ${z}^{d}+c$ with $d>2$. We also show that the conjecture does not hold for ${\mathrm{e}}^{2\pi i\theta }\left(z+{z}^{d}\right)$ with $d>2$.
Nous donnons une nouvelle preuve de la conjecture suivante de Yoccoz  :$\left(\exists C\in ℝ\right)\phantom{\rule{3.33333pt}{0ex}}\left(\forall \theta \in ℝ\setminus ℚ\right)\phantom{\rule{1em}{0ex}}log\mathrm{rad}\phantom{\rule{0.166667em}{0ex}}\Delta \left({Q}_{\theta }\right)\le -Y\left(\theta \right)+C,$${Q}_{\theta }\left(z\right)={\mathrm{e}}^{2\pi i\theta }z+{z}^{2}$, $\Delta \left({Q}_{\theta }\right)$ est son disque de Siegel si ${Q}_{\theta }$ est linéarisable (ou $\varnothing$ sinon), $\mathrm{rad}\phantom{\rule{0.166667em}{0ex}}\Delta \left({Q}_{\theta }\right)$ est le rayon conforme du disque de Siegel de ${Q}_{\theta }$ (ou $0$ s’il n’y en a pas) et $Y\left(\theta \right)$ 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 ${z}^{d}+c$ avec $d>2$. Nous montrons également que la conjecture ne tient pas pour ${\mathrm{e}}^{2\pi i\theta }\left(z+{z}^{d}\right)$ avec $d>2$.
DOI : https://doi.org/10.5802/aif.2603
Classification:  37F50
Keywords: Siegel disks, quadratic polynomials, harmonic and subharbonic functions, conformal radius, holomorphic motions
@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},
publisher = {Association des Annales de l'institut Fourier},
volume = {61},
number = {1},
year = {2011},
pages = {319-350},
doi = {10.5802/aif.2603},
zbl = {1223.37061},
mrnumber = {2828132},
language = {en},
url = {https://aif.centre-mersenne.org/item/AIF_2011__61_1_319_0}
}

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. https://aif.centre-mersenne.org/item/AIF_2011__61_1_319_0/

[1] Bers, Lipman; Royden, H. L. Holomorphic families of injections, Acta Math., Tome 157 (1986) no. 3-4, pp. 259-286 | Article | MR 857675 | Zbl 0619.30027

[2] Buff, Xavier Virtually repelling fixed points, Publ. Mat., Tome 47 (2003) no. 1, pp. 195-209 | MR 1970900 | Zbl 1043.37014

[3] Buff, Xavier; Chéritat, Arnaud Upper bound for the size of quadratic Siegel disks, Invent. Math., Tome 156 (2004) no. 1, pp. 1-24 | Article | MR 2047656 | Zbl 1087.37041

[4] Buff, Xavier; Chéritat, Arnaud The Brjuno function continuously estimates the size of quadratic Siegel disks, Ann. of Math. (2), Tome 164 (2006) no. 1, pp. 265-312 | Article | MR 2233849 | Zbl 1109.37040

[5] Buff, Xavier; Epstein, Adam L. A parabolic Pommerenke-Levin-Yoccoz inequality, Fund. Math., Tome 172 (2002) no. 3, pp. 249-289 | Article | MR 1898687 | Zbl 1115.37323

[6] Chéritat, Arnaud Recherche d’ensembles de Julia de mesure de Lebesgue positive, Université Paris-Sud (2001) (thèse de doctorat)

[7] Douady, Adrien; Hubbard, John Hamal On the dynamics of polynomial-like mappings, Ann. Sci. École Norm. Sup. (4), Tome 18 (1985) no. 2, pp. 287-343 | Numdam | MR 816367 | Zbl 0587.30028

[8] Epstein, A. L. Infinitesimal Thurston rigidity and the Fatou-Shishikura inequality (1991) (Stony Brook IMS Preprint)

[9] Geyer, Lukas Linearization of structurally stable polynomials, Progress in holomorphic dynamics, Longman, Harlow (Pitman Res. Notes Math. Ser.) Tome 387 (1998), pp. 27-30 | MR 1643012 | Zbl 0955.30022

[10] Hardy, G. H.; Wright, E. M. An introduction to the theory of numbers, Oxford University Press, Oxford (2008) (Revised by D. R. Heath-Brown and J. H. Silverman) | MR 2445243 | Zbl 1159.11001

[11] Mañé, R.; Sad, P.; Sullivan, D. On the dynamics of rational maps, Ann. Sci. École Norm. Sup. (4), Tome 16 (1983) no. 2, pp. 193-217 | Numdam | MR 732343 | Zbl 0524.58025

[12] Pommerenke, Christian Univalent functions, Vandenhoeck & Ruprecht, Göttingen (1975) (With a chapter on quadratic differentials by Gerd Jensen, Studia Mathematica/Mathematische Lehrbücher, Band XXV) | MR 507768 | Zbl 0298.30014

[13] Ransford, Thomas Potential theory in the complex plane, Cambridge University Press, Cambridge, London Mathematical Society Student Texts, Tome 28 (1995) | Article | MR 1334766 | Zbl 0828.31001

[14] Risler, Emmanuel Linéarisation des perturbations holomorphes des rotations et applications, Mém. Soc. Math. Fr. (N.S.) (1999) no. 77, pp. viii+102 | Numdam | MR 1779976 | Zbl 0929.37017

[15] Shishikura, Mitsuhiro On the quasiconformal surgery of rational functions, Ann. Sci. École Norm. Sup. (4), Tome 20 (1987) no. 1, pp. 1-29 | Numdam | MR 892140 | Zbl 0621.58030

[16] Slodkowski, Zbigniew Holomorphic motions and polynomial hulls, Proc. Amer. Math. Soc., Tome 111 (1991) no. 2, pp. 347-355 | Article | MR 1037218 | Zbl 0741.32009

[17] Sullivan, Dennis P.; Thurston, William P. Extending holomorphic motions, Acta Math., Tome 157 (1986) no. 3-4, pp. 243-257 | Article | MR 857674 | Zbl 0619.30026

[18] Yoccoz, Jean-Christophe Théorème de Siegel, nombres de Bruno et polynômes quadratiques, Astérisque (1995) no. 231, pp. 3-88 (Petits diviseurs en dimension $1$) | MR 1367353