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On critical circle homeomorphisms

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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.

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References

  1. Golumbic, M.C.,Algorithmic graph theory and perfect graphs by Academic Press, New York (1980).

    Google Scholar 

  2. Graczyk, J. & Świ atek, G.,Critical circle maps near bifurcation, Commun. Math. Phys.176: (1996), 227–260.

    Google Scholar 

  3. Herman, M.,Conjugaison quasi symétrique des homéomorphismes analytiques du cercle à des rotations, manuscript (1987).

  4. Herman, M.,Are there critical points on the boundaries of singular domains?, Comm. Math. Phys.99: (1985), 593–612.

    Google Scholar 

  5. Herman, M.,Uniformité de la distorsion de Świ atek pour les familles compactes de produits Blaschke, manuscript (1988).

  6. Hu, J. & Sullivan, D.,Topological conjugacy of circle diffeomorphisms, Ergod. Th. and Dyn. Sys.17: (1997), 173–186.

    Google Scholar 

  7. Lehto, O. & Virtanen, K.I.,Quasiconformal mappings in the plane, 2-nd Ed., Springer-Verlag, New York (1973).

    Google Scholar 

  8. Sullivan, D.,Bounds, quadratic differentials and renormalization conjectures, inMathematics in the twenty-first century, American Mathematical Society, Providence, RI (1991).

    Google Scholar 

  9. Świ atek, G.,Rational rotation numbers form maps of the circle, Commun. Math. Phys.119: (1988), 109–128.

    Google Scholar 

  10. Świ atek, G.,Circle homeomorphisms with flat critical points, Fundamenta Mathematicae,138: (1991).

  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.

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Authors and Affiliations

  1. Department of Mathematics, Pennsylvania State University, 16802, University Park, PA, USA

    Grzegorz Światek

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  1. Grzegorz Światek
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Dedicated to the memory of R. Mañé

Partially supported by NSF grant DMS-9704368 and the Sloan Foundation.

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Światek, G. On critical circle homeomorphisms. Bol. Soc. Bras. Mat 29, 329–351 (1998). https://doi.org/10.1007/BF01237654

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  • DOI: https://doi.org/10.1007/BF01237654

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