The Brjuno function continuously estimates the size of quadratic Siegel disks

Xavier Buff¹ ; Arnaud Chéritat²

¹ Institut de Mathématiques de Toulouse, Université Paul Sabatier, Laboratoire Emile Picard, 31062 Toulouse, France
² nstitut de Mathématiques de Toulouse, Université Paul Sabatier, Laboratoire Emile Picard, 31062 Toulouse, France

Annals of mathematics, Tome 164 (2006) no. 1, pp. 265-312.

Résumé

If $\alpha$ is an irrational number, Yoccoz defined the Brjuno function $\Phi$ by \[\Phi(\alpha)=\sum_{n\geq 0} \alpha_0\alpha_1\cdots\alpha_{n-1}\log\frac{1}{\alpha_n},\] where $\alpha_0$ is the fractional part of $\alpha$ and $\alpha_{n+1}$ is the fractional part of ${1/\alpha_n}$. The numbers $\alpha$ such that $\Phi(\alpha)<+\infty$ are called the Brjuno numbers. The quadratic polynomial $P_\alpha:z\mapsto e^{2i\pi \alpha}z+z^2$ has an indifferent fixed point at the origin. If $P_\alpha$ is linearizable, we let $r(\alpha)$ be the conformal radius of the Siegel disk and we set $r(\alpha)=0$ otherwise. Yoccoz [Y] proved that $\Phi(\alpha)=+\infty$ if and only if $r(\alpha)=0$ and that the restriction of $\alpha\mapsto \Phi(\alpha)+\log r(\alpha)$ to the set of Brjuno numbers is bounded from below by a universal constant. In [BC2], we proved that it is also bounded from above by a universal constant. In fact, Marmi, Moussa and Yoccoz [MMY] conjecture that this function extends to $\mathbb{R}$ as a Hölder function of exponent $1/2$. In this article, we prove that there is a continuous extension to $\mathbb{R}$.

Détail

DOI : 10.4007/annals.2006.164.265