Upper bound for the size of quadratic Siegel disks

If is an irrational number, we let {p_n/q_n}_n0, be the approximants given by its continued fraction expansion. The Bruno series B() is defined as The quadratic polynomial P:ze²ⁱz+z² has an indifferent fixed point at the origin. If P is linearizable, we let r() be the conformal radius of the Siegel disk and we set r()=0 otherwise. Yoccoz proved that if B()=, then r()=0 and P 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 with B()<, we have Together with former results of Yoccoz (see [Y]), this proves the conjectured boundedness of B()+logr().