The Brjuno function continuously estimates the size of quadratic Siegel disks
成果类型：
Article
署名作者：
Buff, Xavier; Cheritiat, Arnaud
刊物名称：
ANNALS OF MATHEMATICS
ISSN/ISSBN：
0003-486X
DOI：
10.4007/annals.2006.164.265
发表日期：
2006
页码：
265-312
关键词：
摘要：
If a is an irrational number, Yoccoz defined the Brjuno function Phi by Phi(alpha) = Sigma(n >= 0) alpha(0)alpha(1)...alpha(n-1)log 1/alpha(n), where alpha(0) is the fractional part of alpha and alpha(n+1) is the fractional part of 1/alpha(n). The numbers alpha such that Phi(alpha) < +infinity are called the Brjuno numbers. The quadratic polynomial P-alpha : z negated right arrow e(2i pi alpha)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 [Y] proved that Phi(alpha) = +infinity if and only if r(alpha) = 0 and that the restriction of alpha negated right arrow Phi(alpha) + logr(alpha) to the set of Brjuno numbers is bounded from below by a universal constant. In [BC2], we proved that it is also bounded from above by a universal constant. In fact, Marmi, Moussa and Yoccoz [MMY] conjecture that this function extends to R as a Holder function of exponent 1/2. In this article, we prove that there is a continuous extension to R.