On the Size of Quadratic Siegel Disks: Part I

Abstract

If $\a$ is an irrational number, we let $\{p_n/q_n\}_{n\geq 0}$, be the approximants given by its continued fraction expansion. The Bruno series $B(\a)$ is defined as $$B(\a)=\sum_{n\geq 0} \frac{\log q_{n+1}}{q_n}.$$ The quadratic polynomial $P_\a:z\mapsto e^{2iπ\a}z+z^2$ has an indifferent fixed point at the origin. If $P_\a$ is linearizable, we let $r(\a)$ be the conformal radius of the Siegel disk and we set $r(\a)=0$ otherwise. Yoccoz proved that if $B(\a)=\infty$, then $r(\a)=0$ and $P_\a$ is not linearizable. In this article, we present a different proof and we show that there exists a constant $C$ such that for all irrational number $\a$ with $B(\a)<\infty$, we have $$B(\a)+\log r(\a) < C.$$ Together with former results of Yoccoz (see \cite{y}), this proves the conjectured boundedness of $B(\a)+\log r(\a)$.