Upper bound for the size of quadratic Siegel disks

被引：19

作者：

Buff, X ^{[1
]}

Chéritat, A ^{[1
]}

机构：

[1] Univ Toulouse 3, Lab Emile Picard, F-31062 Toulouse, France

来源：

INVENTIONES MATHEMATICAE | 2004年 / 156卷 / 01期

关键词：

Continue Fraction; Irrational Number; Continue Fraction Expansion; Siegel Disk; Conformal Radius;

D O I：

10.1007/s00222-003-0331-6

中图分类号：

O1 [数学];

学科分类号：

0701 ; 070101 ;

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).

引用

页码：1 / 24

页数：24