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
The quadratic polynomial P α:z↦e 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
Together with former results of Yoccoz (see [Y]), this proves the conjectured boundedness of B(α)+logr(α).
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Ahlfors, L.: Conformal Invariants. McGraw-Hill Series in Higher Math.
Bruno, A.D.: Analytic forms of differential equations. Trans. Mosc. Math. Soc. 25 (1971)
Buff, X., Chéritat, A.: Quadratic Siegel disks with smooth boundaries. Preprint, 242, Toulouse (2002)
Chéritat, A.: Recherche d’ensembles de Julia de mesure de Lebesgue positive. Thèse, Université de Paris-Sud, Orsay (2001)
Douady, A.: Prolongement de mouvements holomorphes [d’après Slodkowski et autres]. Séminaire Bourbaki 775 (1993)
Douady, A., Hubbard, J.H.: Étude dynamique des polynômes complexes I & II. Publ. Math. Orsay (1984–85)
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)
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
Milnor, J.: Dynamics in one complex variable, Introductory Lectures. Braunschweig: Friedr. Vieweg & Sohn 1999
Petersen, C.L.: On the Pommerenke, Levin Yoccoz inequality. Ergodic Theory Dyn. Syst. 13, 785–806 (1993)
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)
Siegel, C.L.: Iteration of analytic functions. Ann. Math. 43 (1942)
Slodkowski, Z.: Extensions of holomorphic motions. Prépublication IHES/M/92/96 (1993)
Yoccoz, J.C.: Petits diviseurs en dimension 1. Astérisque 231 (1995)
Author information
Authors and Affiliations
Corresponding authors
Rights and permissions
About this article
Cite this article
Buff, X., Chéritat, A. Upper bound for the size of quadratic Siegel disks. Invent. math. 156, 1–24 (2004). https://doi.org/10.1007/s00222-003-0331-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00222-003-0331-6