Upper bound for the size of quadratic Siegel disks
成果类型：
Article
署名作者：
Buff, X; Chéritat, A
署名单位：
Universite de Toulouse; Universite Toulouse III - Paul Sabatier
刊物名称：
INVENTIONES MATHEMATICAE
ISSN/ISSBN：
0020-9910
DOI：
10.1007/s00222-003-0331-6
发表日期：
2004
页码：
1-24
关键词：
摘要：
If alpha is an irrational number, we let {p(n)/q(n)}(ngreater than or equal to0), be the approximants given by its continued fraction expansion. The Bruno series B(alpha) is defined as [GRAPHICS] The quadratic polynomial P-alpha : z \--> l(2ipipialpha) z + z(2) has an indifferent fixed point at the origin. If P-alpha is linearizable, we let r(alpha) be the conformal radius of the Siegel disk and we set r(alpha) = 0 otherwise. Yoccoz proved that if B(alpha) = infinity, then r(alpha) = 0 and P-alpha 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(alpha) < infinity, we have B(alpha) + log r(alpha) < C. Together with former results of Yoccoz (see [Y]), this proves the conjectured boundedness of B(alpha) + log r(alpha).