A new proof of a conjecture of Yoccoz
Annales de l'Institut Fourier, Volume 61 (2011) no. 1, pp. 319-350.

We give a new proof of the following conjecture of Yoccoz:

(C)(θ)logradΔ(Qθ)-Y(θ)+C,

where Q θ (z)=e 2πiθ z+z 2 , Δ(Q θ ) is its Siegel disk if Q θ is linearizable (or otherwise), radΔ(Q θ ) is the conformal radius of the Siegel disk of Q θ (or 0 if there is none) and Y(θ) 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 e 2πiθ (z+z d ) with d>2.

Nous donnons une nouvelle preuve de la conjecture suivante de Yoccoz  :

(C)(θ)logradΔ(Qθ)-Y(θ)+C,

Q θ (z)=e 2πiθ z+z 2 , Δ(Q θ ) est son disque de Siegel si Q θ est linéarisable (ou sinon), radΔ(Q θ ) est le rayon conforme du disque de Siegel de Q θ (ou 0 s’il n’y en a pas) et Y(θ) 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 e 2πiθ (z+z d ) avec d>2.

DOI: 10.5802/aif.2603
Classification: 37F50
Keywords: Siegel disks, quadratic polynomials, harmonic and subharbonic functions, conformal radius, holomorphic motions
Buff, Xavier 1; Chéritat, Arnaud 2

1 Université Paul Sabatier Institut de Mathématiques de Toulouse 118, route de Narbonne 31062 Toulouse Cedex 9 (France)
2 C.N.R.S Université Paul Sabatier Institut de Mathématiques de Toulouse 118, route de Narbonne 31062 Toulouse Cedex 9
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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. https://aif.centre-mersenne.org/articles/10.5802/aif.2603/

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