Duke Mathematical Journal

Siegel disks with critical points in their boundaries

Jacek Graczyk and Grzegorz Świątek

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


Consider an analytic function f which has a Siegel disk properly contained in the domain of holomorphy. We prove that if the rotation number is of bounded type, then f has a critical point in the boundary of the Siegel disk.

Article information

Duke Math. J., Volume 119, Number 1 (2003), 189-196.

First available in Project Euclid: 23 April 2004

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 37Fxx: Complex dynamical systems [See also 30D05, 32H50]
Secondary: 30Cxx: Geometric function theory


Graczyk, Jacek; Świątek, Grzegorz. Siegel disks with critical points in their boundaries. Duke Math. J. 119 (2003), no. 1, 189--196. doi:10.1215/S0012-7094-03-11916-X. https://projecteuclid.org/euclid.dmj/1082744709

Export citation


  • L. Ahlfors and G. Weill, A uniqueness theorem for Beltrami equations, Proc. Amer. Math. Soc. 13 (1962), 975--978.
  • A. Douady, Disques de Siegel et anneaux de Herman, Astérisque 152 --.153 (1987), 4, 151--172., Séminaire Bourbaki 1986/87, exp. 677.
  • P. Fatou, Sur les équations fonctionnelles, Bull. Soc. Math. France 47 (1919), 161--271.; 48 (1920), 33--94., 208--314.
  • É. Ghys, Transformations holomorphes au voisinage d'une courbe de Jordan, C. R. Acad. Sci. Paris Sér. I Math. 298 (1984), 385--388.
  • M.-R. Herman, Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations, Inst. Hautes Études Sci. Publ. Math. 49 (1979), 5--233.
  • --. --. --. --., Are there critical points on the boundaries of singular domains?, Comm. Math. Phys. 99 (1985), 593--612.
  • --. --. --. --., ``Recent results and some open questions on Siegel's linearization theorem of germs of complex analytic diffeomorphisms of $C\sp n$ near a fixed point'' in VIIIth International Congress on Mathematical Physics (Marseille, 1986), ed. M. Mebkhout and R. Sénéor, World Sci., Singapore, 1987, 138--184.
  • C. L. Siegel, Iterations of analytic functions, Ann. of Math. (2) 43 (1942), 607--612.