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How regular can the boundary of a quadratic Siegel disk be?


Authors: Xavier Buff and Arnaud Chéritat
Journal: Proc. Amer. Math. Soc. 135 (2007), 1073-1080
MSC (2000): Primary 37F50, 37F10, 46B50
DOI: https://doi.org/10.1090/S0002-9939-06-08578-9
Published electronically: September 26, 2006
MathSciNet review: 2262908
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Abstract: In the family of quadratic polynomials with an irrationally indifferent fixed point, we show the existence of Siegel disks with a fine control on the degree of regularity of the linearizing map on their boundary. A general theorem is stated and proved. As a particular case, we show that in the quadratic family, there are Siegel disks whose boundaries are $ C^n$ but not $ C^{n+1}$ Jordan curves.


References [Enhancements On Off] (What's this?)

  • [A] A. AVILA, Smooth Siegel disks via semicontinuity: A remark on a proof of Buff and Cheritat, math.DS/0305272
  • [ABC] A. AVILA, X. BUFF $ \&$ A. CHÉRITAT, Siegel disks with smooth boundaries, Acta Mathematica 193, 1-30 (2004). MR 2155030 (2006e:37073)
  • [BC] X. BUFF $ \&$ A. CHÉRITAT, Quadratic Siegel disks with smooth boundaries, preprint no. 242 at Université Paul Sabatier, Toulouse, France (2002).
  • [G] L. GEYER Smooth Siegel discs without number theory: A remark on a proof by Buff and Chéritat, submitted (2003).
  • [H1] M.R. HERMAN, Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations, Publ. Math. I.H.E.S. 49, 5-233 (1979). MR 0538680 (81h:58039)
  • [H2] M.R. HERMAN, Are there critical points on the boundaries of singular domains?, Comm. Math. Phys. 99, 593-612 (1985). MR 0796014 (86j:58067)
  • [M] J. MILNOR, Dynamics in one complex variable, Introductory Lectures, Friedr. Vieweg $ \&$ Sohn, Braunschweig (1999). MR 1721240 (2002i:37057)
  • [PM1] R. P´EREZ-MARCO, Siegel disks with smooth boundary, Preprint (1997).
  • [PM2] R. P´EREZ-MARCO, Siegel disks with quasi-analytic boundary, Preprint no. 52 (1997) at Université Paris-Sud.
  • [Po] C. POMMERENKE, Boundary Behavior of Conformal Maps, Grundlehren der mathematischen Wissenschaften 299, Springer-Verlag (1992). MR 1217706 (95b:30008)
  • [PZ] C.L. PETERSEN $ \&$ S. ZAKERI, On the Julia Set of a Typical Quadratic Polynomial with a Siegel disk, Ann. of Math. (2) 159 (2004), no. 1, 1-52. MR 2051390 (2005c:37085)
  • [Ro] B. RODIN, Intrinsic rotations of simply connected regions, Complex Variables Theory Appl. 2 (1984), no. 3-4, 319-326. MR 0743955 (85m:30026)
  • [Ru] W. RUDIN, Functional Analysis, McGraw-Hill Series in Higher Math., second edition (1991). MR 1157815 (92k:46001)

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Additional Information

Xavier Buff
Affiliation: Université Paul Sabatier, Laboratoire Emile Picard, 118, route de Narbonne, 31062 Toulouse Cedex, France
Email: buff@picard.ups-tlse.fr

Arnaud Chéritat
Affiliation: Université Paul Sabatier, Laboratoire Emile Picard, 118, route de Narbonne, 31062 Toulouse Cedex, France
Email: cheritat@picard.ups-tlse.fr

DOI: https://doi.org/10.1090/S0002-9939-06-08578-9
Received by editor(s): January 28, 2005
Received by editor(s) in revised form: November 2, 2005
Published electronically: September 26, 2006
Communicated by: Linda Keen
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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