Skip to main content
SpringerLink
Log in

Conformally natural extensions of continuous circle maps: II. The general case

  • Published:
Journal d'Analyse Mathématique Aims and scope

Abstract

We introduce a procedure for extending continuous circle maps in a conformally natural way to continuous maps from the closed disk, bounded by the circle to itself which generalizes Douady-Earle’s method of constructing conformally natural extensions of circle homeomorphisms. We also provide a criterion for the extensions to be surjective maps from the closed disk to itself.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. W. Abikoff, Conformal barycenters and the Douady-Earle extension-a discrete dynamical approach, J. Anal. Math. 86 (2002), 221–234.

    Article  MathSciNet  MATH  Google Scholar 

  2. W. Abikoff, C. Earle, and S. Mitra, Barycentric extensions of monotone maps of the circle, In the Tradition of Alhfors and Bers III, Contemp. Math. 355, Amer. Math. Soc., Providence, RI, 2004, pp. 1–20.

    Chapter  Google Scholar 

  3. W. Abikoff and T. Ye, Computing the Douady-Earle extension, Lipa’s Legacy, Contemp. Math. 211, Amer. Math. Soc., Providence, RI, 1997, pp. 1–8.

    Google Scholar 

  4. A. Beurling and L. V. Ahlfors, The boundary correspondence under quasiconformal mappings, Acta Math. 96 (1956), 125–142.

    Article  MathSciNet  MATH  Google Scholar 

  5. G. Cui, Integrably asymptotic affine homeomorphisms of the circle and Teichmüller spaces, Sci. China Ser. A 43 (2000), 267–279.

    Article  MathSciNet  MATH  Google Scholar 

  6. G. Cui and M. Zinsmeister, BMO-Teichmüller spaces, Illinois J. Math. 48 (2004), 1223–1233.

    MathSciNet  MATH  Google Scholar 

  7. A. Douady and C. J. Earle, Conformally natural extension of homeomorphisms of circle, Acta Math. 157 (1986), 23–48.

    Article  MathSciNet  MATH  Google Scholar 

  8. C. J. Earle, V. Markovic, and D. Šarić, Barycentric extension and the Bers embedding for asymptotic Teichmüller space, Complex Manifolds and Hyperbolic Geometry, Contemp. Math. 311, Amer. Math. Soc., Providence RI, 2002, pp. 87–106.

    Chapter  Google Scholar 

  9. J. Fan and J. Hu, Topological characterization of an asymptotic Teichmüller space through measured geodesic laminations, Pure Appl. Math. Q. 11 (2015), 403–449.

    Article  MathSciNet  MATH  Google Scholar 

  10. J. Fan and J. Hu, Characterization of the asymptotic Teichmüller space of the open unit disk through shears, Pure Appl, Math. Q. 10 (2014), 513–546.

    Article  MathSciNet  MATH  Google Scholar 

  11. J. Fan and J. Hu, Holomorphic contractibility and other properties of the Weil-Petersson and VMOA Teichmüller spaces, Ann. Acad. Sci. Fenn. 41 (2016), 587–600.

    Article  MathSciNet  MATH  Google Scholar 

  12. J. Hu and O. Muzician, Cross-ratio distortion and Douady-Earle extension: I. A new upper bound on quasiconformality, J. London Math. Soc. (2) 86 (2012), 387–406.

    Article  MathSciNet  MATH  Google Scholar 

  13. J. Hu and O. Muzician, Cross-ratio distortion and Douady-Earle extension: II. Quasiconformality and asymptotic conformality are local, J. Anal. Math. 117 (2012), 249–271.

    Article  MathSciNet  MATH  Google Scholar 

  14. J. Hu and O. Muzician, Conformally natural extensions of continuous circle maps: I. The case when the pushforward measure has no atom, Mapping Riemann Spaces and Teichmüller Spaces, Contemp. Math. 575, Amer. Math. Soc., Providence, RI, 2012, pp. 171–198.

    Google Scholar 

  15. J. Hu and S. Pal, Boundary differentiability of Douady-Earle extensions of diffeomorphisms of Sn, Pure and Appl. Math. Q., to appear.

  16. V. Markovic, Quasisymmetric groups, J. Amer. Math. Soc. 19 (2006), 673–715.

    Article  MathSciNet  MATH  Google Scholar 

  17. J. W. Milnor, Topology from the Differentiable Viewpoint, University Virginia Press, 1965.

    MATH  Google Scholar 

  18. H. Miyachi and D. Šarić, Uniform weak* topology and earthquake in the hyperbolic plane, Proc. London Math. Soc. (3) 105 (2012), 1123–1148.

    Article  MathSciNet  MATH  Google Scholar 

  19. O. Muzician, Conformally natural extensions of continuous circle maps, Ph. D. thesis, Graduate Center of CUNY, 2012.

    MATH  Google Scholar 

  20. C. L. Petersen, Conformally natural extensions revisited, arXiv:1102.1470[math.GT].

  21. D. Šarić, Circle homeomorphisms and shears, Geom. Topol. 14 (2010), 2405–2430.

    Article  MathSciNet  MATH  Google Scholar 

  22. Y. Shen and H. Wei, Universal Teichmüller space and BMO, Adv. Math. 234 (2013), 129–148.

    Article  MathSciNet  MATH  Google Scholar 

  23. W. P. Thurston, Earthquakes in two-dimensional hyperbolic geometry in Low-dimensional Topology and Kleinian Groups, Cambridge University Press, Cambridge, 1986, pp. 91–112.

    Google Scholar 

  24. P. Tukia, Quasiconformal extension of quasisymmetric mappings compatible with a Möbius group, Acta Math. 154 (1985), 153–193.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Brooklyn College of CUNY, Brooklyn, NY, 11210, USA

    Jun Hu

  2. Ph.D. Program in Mathematics, Graduate Center of CUNY, 365 Fifth Avenue, New York, NY, 10016, USA

    Jun Hu

  3. Department of Mathematics, Borough of Manhattan Community College of CUNY, 199 Chambers Street, New York, NY, 10007, USA

    Oleg Muzician

Authors
  1. Jun Hu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Oleg Muzician
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jun Hu.

Additional information

The research was partially supported by PSC-CUNY research awards.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Hu, J., Muzician, O. Conformally natural extensions of continuous circle maps: II. The general case. JAMA 132, 81–107 (2017). https://doi.org/10.1007/s11854-017-0014-7

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11854-017-0014-7

Search

Navigation