On the Size of Quadratic Siegel Disks: Part I

If $\text{a}$ is an irrational number, we let $\{p_n\mathrm{/}{q}_n{\}}_{n\ge 0}$, be the approximants given by its continued fraction expansion. The Bruno series $B(\text{a})$ is defined as $$B(\text{a})=\sum _{n\ge 0}\frac{\log q_{n+1}}{q_n}\mathrm{.}$$ The quadratic polynomial $P_{\text{a}}:z\mapsto e^{2i\pi \text{a}}z+z^2$ has an indifferent fixed point at the origin. If $P_{\text{a}}$ is linearizable, we let $r(\text{a})$ be the conformal radius of the Siegel disk and we set $r(\text{a})=0$ otherwise. Yoccoz proved that if $B(\text{a})=\mathrm{\infty }$, then $r(\text{a})=0$ and $P_{\text{a}}$ 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 $\text{a}$ with $B(\text{a})<\mathrm{\infty }$, we have $$B(\text{a})+\log r(\text{a})<C\mathrm{.}$$ Together with former results of Yoccoz (see \cite{y}), this proves the conjectured boundedness of $B(\text{a})+\log r(\text{a})$.