Upper bound for the size of quadratic Siegel disks
Upper bound for the size of quadratic Siegel disks
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DOI：
10.1007/s00222-003-0331-6
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发表时间：
2004-04
影响因子：
3.1
通讯作者：
Xavier Buff;Arnaud Chéritat
中科院分区：
文献类型：
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作者：
Xavier Buff;Arnaud Chéritat
若α是无理数，则{pn/qn}n≥0，是由其连分式展开式给出的逼近式. Bruno级数B（α）定义为二次多项式P α：z ∈ 2 i παz+ z2在原点有一个中立不动点.如果P α是可线性化的，我们设r（α）为Siegel圆盘的共形半径，否则设r（α）=0。Yoccoz证明了当B（α）=∞时，r（α）=0，P α不可线性化。本文给出了一个不同的证明，证明了存在一个常数C，使得对任意无理数α，当B（α）<∞时，我们有.结合Yoccoz的结果（见[Y]），证明了B（α）+logr（α）的有界性.
If α is an irrational number, we let {pn/qn}n≥0, be the approximants given by its continued fraction expansion. The Bruno seriesB(α) is defined asThe quadratic polynomialPα:z↦e2iπαz+z2has an indifferent fixed point at the origin. IfPαis linearizable, we letr(α) be the conformal radius of the Siegel disk and we setr(α)=0 otherwise. Yoccoz proved that ifB(α)=∞, thenr(α)=0 andPαis not linearizable. In this article, we present a different proof and we show that there exists a constantCsuch that for all irrational number α withB(α)<∞, we haveTogether with former results of Yoccoz (see [Y]), this proves the conjectured boundedness ofB(α)+logr(α).