Resumen de The Brjuno function continuously estimates the size of quadratic Siegel disks
If ¿Á is an irrational number, Yoccoz defined the Brjuno function ¿µ by ¿µ(¿Á) =n¡Ý0 ¿Á0¿Á1 ¡¿ ¡¿ ¡¿ ¿Án.1 log 1 ¿Án , where ¿Á0 is the fractional part of ¿Á and ¿Án+1 is the fractional part of 1/¿Án. The numbers ¿Á such that ¿µ(¿Á) < +¡Þ are called the Brjuno numbers. The quadratic polynomial P¿Á : z ¡ú e2i¿Ð¿Áz + z2 has an indifferent fixed point at the origin. If P¿Á is linearizable, we let r(¿Á) be the conformal radius of the Siegel disk and we set r(¿Á) = 0 otherwise. Yoccoz [Y] proved that ¿µ(¿Á) = +¡Þ if and only if r(¿Á) = 0 and that the restriction of ¿Á ¡ú ¿µ(¿Á) + log r(¿Á) 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 R as a H¡§older function of exponent 1/2. In this article, we prove that there is a continuous extension to R.
