On critical circle homeomorphisms

  • Grzegorz Światek
Article

Abstract

We prove that an analytic circle homeomorphism without periodic orbits is conjugated to the linear rotation by a quasi-symmetric map if an only if its rotation number is of constant type. Next, we consider automorphisms of quasi-conformal Jordan curves, without periodic orbits and holomorphic in a neighborhood. We prove a “Denjoy theorem” that such maps are conjugated to a rotation on the circle.

Keywords

Cross-ratio inequality quasi-symmetric conjugacy Denjoy theorem 

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References

  1. [1]
    Golumbic, M.C.,Algorithmic graph theory and perfect graphs by Academic Press, New York (1980).Google Scholar
  2. [2]
    Graczyk, J. & Świ atek, G.,Critical circle maps near bifurcation, Commun. Math. Phys.176: (1996), 227–260.Google Scholar
  3. [3]
    Herman, M.,Conjugaison quasi symétrique des homéomorphismes analytiques du cercle à des rotations, manuscript (1987).Google Scholar
  4. [4]
    Herman, M.,Are there critical points on the boundaries of singular domains?, Comm. Math. Phys.99: (1985), 593–612.Google Scholar
  5. [5]
    Herman, M.,Uniformité de la distorsion de Świ atek pour les familles compactes de produits Blaschke, manuscript (1988).Google Scholar
  6. [6]
    Hu, J. & Sullivan, D.,Topological conjugacy of circle diffeomorphisms, Ergod. Th. and Dyn. Sys.17: (1997), 173–186.Google Scholar
  7. [7]
    Lehto, O. & Virtanen, K.I.,Quasiconformal mappings in the plane, 2-nd Ed., Springer-Verlag, New York (1973).Google Scholar
  8. [8]
    Sullivan, D.,Bounds, quadratic differentials and renormalization conjectures, inMathematics in the twenty-first century, American Mathematical Society, Providence, RI (1991).Google Scholar
  9. [9]
    Świ atek, G.,Rational rotation numbers form maps of the circle, Commun. Math. Phys.119: (1988), 109–128.Google Scholar
  10. [10]
    Świ atek, G.,Circle homeomorphisms with flat critical points, Fundamenta Mathematicae,138: (1991).Google Scholar
  11. [11]
    Yoccoz, J.-C.,Il n'y a pas de contre-exemple de Denjoy analitique, C. R. Acad. Sci. Paris,298: (1984), Serue I, no. 7.Google Scholar

Copyright information

© Sociedade Brasileira de Matemática 1998

Authors and Affiliations

  • Grzegorz Światek
    • 1
  1. 1.Department of MathematicsPennsylvania State UniversityUniversity ParkUSA

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