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Inventiones mathematicae

, Volume 156, Issue 1, pp 1–24 | Cite as

Upper bound for the size of quadratic Siegel disks

  • Xavier Buff
  • Arnaud Chéritat
Article

Abstract

If α is an irrational number, we let {p n /q n } n≥0, be the approximants given by its continued fraction expansion. The Bruno series B(α) is defined as
$$B(\alpha)=\sum_{n\geq 0} \frac{\log q_{n+1}}{q_n}.$$
The quadratic polynomial P α:ze 2iπα z+z 2 has an indifferent fixed point at the origin. If P α 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
$$B(\alpha)+\log r(\alpha) < C.$$
Together with former results of Yoccoz (see [Y]), this proves the conjectured boundedness of B(α)+logr(α).

Keywords

Continue Fraction Irrational Number Continue Fraction Expansion Siegel Disk Conformal Radius 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag 2003

Authors and Affiliations

  1. 1.Laboratoire Emile PicardUniversité Paul SabatierToulouseFrance

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