Upper bound for the size of quadratic Siegel disks

Abstract

If α is an irrational number, we let {p n /q n } n≥0, be the approximants given by its continued fraction expansion. The Bruno series B(α) is defined as $$B(\alpha)=\sum_{n\geq 0} \frac{\log q_{n+1}}{q_n}.$$ The quadratic polynomial P α:z↦e 2iπα z+z 2 has an indifferent fixed point at the origin. If P α is linearizable, we let r(α) be the conformal radius of the Siegel disk and we set r(α)=0 otherwise. Yoccoz proved that if B(α)=∞, then r(α)=0 and P α 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 α with B(α)<∞, we have $$B(\alpha)+\log r(\alpha) < C.$$ Together with former results of Yoccoz (see [Y]), this proves the conjectured boundedness of B(α)+logr(α).