Upper bound for the size of quadratic Siegel disks

Abstract

If α is an irrational number, we let {pn/q n}n≥0, be the approximants given by its continued fraction expansion. The Bruno series B(α) is defined as B(α) = ∑n≥0/log qn+1/qn. The quadratic polynomial Pα : z → e2iπαz + z2 has an indifferent fixed point at the origin. If Pα 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(α) + log r(α) < C. Together with former results of Yoccoz (see [Y]), this proves the conjectured boundedness of B(α) + log r(α).