The Brjuno function continuously estimates the size of quadratic Siegel disks

Abstract

If alpha is an irrational number, we define Yoccoz's Brjuno function Phi by Phi(alpha)=sum_{n geq 0} alpha_0*alpha_1*...*alpha_{n-1}*log(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 -> 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 proved that Phi(alpha)=infty if and only if r(alpha)=0 and that the restriction of alpha -> Phi(alpha)+log r(alpha) to the set of Brjuno numbers is bounded from below by a universal constant. We proved that it is also bounded from above by a universal constant. In fact, Marmi, Moussa and Yoccoz conjecture that this function extends to $R$ as a H\"older function of exponent 1/2. In this article, we prove that there is a continuous extension to $R$.