Inventiones mathematicae

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

Upper bound for the size of quadratic Siegel disks

  • Xavier Buff
  • Arnaud Chéritat


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).


Continue Fraction Irrational Number Continue Fraction Expansion Siegel Disk Conformal Radius 
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  1. 1.
    Ahlfors, L.: Conformal Invariants. McGraw-Hill Series in Higher Math. Google Scholar
  2. 2.
    Bruno, A.D.: Analytic forms of differential equations. Trans. Mosc. Math. Soc. 25 (1971) Google Scholar
  3. 3.
    Buff, X., Chéritat, A.: Quadratic Siegel disks with smooth boundaries. Preprint, 242, Toulouse (2002) Google Scholar
  4. 4.
    Chéritat, A.: Recherche d’ensembles de Julia de mesure de Lebesgue positive. Thèse, Université de Paris-Sud, Orsay (2001) Google Scholar
  5. 5.
    Douady, A.: Prolongement de mouvements holomorphes [d’après Slodkowski et autres]. Séminaire Bourbaki 775 (1993) Google Scholar
  6. 6.
    Douady, A., Hubbard, J.H.: Étude dynamique des polynômes complexes I & II. Publ. Math. Orsay (1984–85) Google Scholar
  7. 7.
    Goldberg, L.R., Milnor, J.: Fixed points of polynomial maps. Part II. Fixed point portraits. Ann. Sci. Éc. Norm. Supér., IV. Sér. 26, 51–98 (1993) Google Scholar
  8. 8.
    Hubbard, J.H.: Local connectivity of Julia sets and bifurcation loci: three theorems of J.C. Yoccoz. In: Topological Methods in Modern Mathematics, ed. by L.R. Goldberg and A.V. Phillips, pp. 467–511. Publish or Perish 1993 Google Scholar
  9. 9.
    Milnor, J.: Dynamics in one complex variable, Introductory Lectures. Braunschweig: Friedr. Vieweg & Sohn 1999 Google Scholar
  10. 10.
    Petersen, C.L.: On the Pommerenke, Levin Yoccoz inequality. Ergodic Theory Dyn. Syst. 13, 785–806 (1993) MathSciNetzbMATHGoogle Scholar
  11. 11.
    Pérez-Marco, R.: Sur les dynamiques holomorphes non linéarisables et une conjecture de V.I. Arnold. Ann. Sci. Éc. Norm. Supér., IV. Sér. 26, 565–644 (1993) Google Scholar
  12. 12.
    Siegel, C.L.: Iteration of analytic functions. Ann. Math. 43 (1942) Google Scholar
  13. 13.
    Slodkowski, Z.: Extensions of holomorphic motions. Prépublication IHES/M/92/96 (1993) Google Scholar
  14. 14.
    Yoccoz, J.C.: Petits diviseurs en dimension 1. Astérisque 231 (1995)Google Scholar

Copyright information

© Springer-Verlag 2003

Authors and Affiliations

  1. 1.Laboratoire Emile PicardUniversité Paul SabatierToulouseFrance

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