The Brjuno function continuously estimates the size of quadratic Siegel disks

Abstract

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 $\mathbb{R}$.

Authors

Xavier Buff

Institut de Mathématiques de Toulouse
Université Paul Sabatier
Laboratoire Emile Picard
31062 Toulouse
France

Arnaud Chéritat

nstitut de Mathématiques de Toulouse
Université Paul Sabatier
Laboratoire Emile Picard
31062 Toulouse
France