Geometriae Dedicata

, Volume 162, Issue 1, pp 283–304 | Cite as

Teichmüller rays and the Gardiner–Masur boundary of Teichmüller space II

  • Hideki Miyachi
Original Paper


In this paper, we study the asymptotic behavior of Teichmüller geodesic rays in the Gardiner–Masur compactification. We will observe that any Teichmüller geodesic ray converges in the Gardiner–Masur compactification. Therefore, we get a mapping from the space of projective measured foliations to the Gardiner–Masur boundary by assigning the limits of associated Teichmüller rays. We will show that this mapping is injective but is neither surjective nor continuous. We also discuss the set of points where this mapping is bicontinuous.


Teichmüller space Teichmüller geodesic rays Gardiner–Masur boundary Thurston boundary 

Mathematics Subject Classification

30F60 32G15 32F45 57M50 


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  1. 1.
    Ahlfors L.V.: Lectures on Quasiconformal Mappings, Van Nostrand Mathematical Studies, vol. 10. D. Van Nostrand Co., Inc., Toronto (1966)Google Scholar
  2. 2.
    Bonahon F.: Bouts des variétés hyperboliques de dimension 3. Ann. Math. 124(1), 71–158 (1986)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Bonahon F.: The geometry of Teichmüller space via geodesic currents. Invent. Math. 92(1), 139–162 (1988)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Bridson M., Haefliger A.: Metric Spaces of Non-positive Curvature. Grundlehren der Mathematischen Wissenschaften, vol. 319. Springer, Berlin (1999)Google Scholar
  5. 5.
    Douady, A., Fathi, A., Fried, D., Laudenbach, F., Poénaru, V., Shub, M.: Travaux de Thurston sur les surfaces. Séminaire Orsay (seconde édition). Astérisque No. 66-67, Société Mathématique de France, Paris (1991)Google Scholar
  6. 6.
    Duchin M., Leininger C.J., Rafi K.: Length spectra and degeneration of flat metrics. Invent. Math. 182, 231–277 (2010)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Gardiner F.: Measured foliations and the minimal norm property for quadratic differentials. Acta Math. 152(1–2), 57–76 (1984)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Gardiner F., Masur H.: Extremal length geometry of Teichmüller space. Complex Variables Theory Appl. 16(2–3), 209–237 (1991)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Hubbard J., Masur H.: Quadratic differentials and foliations. Acta Math. 142(3–4), 221–274 (1979)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Imayoshi Y., Taniguchi M.: Introduction to Teichmüller spaces. Springer, New York (1992)MATHCrossRefGoogle Scholar
  11. 11.
    Ivanov, N.: Isometries of Teichmüller spaces from the point of view of Mostow rigidity. In: Turaev, V., Vershik, A. (eds.), Topology, Ergodic Theory, Real Algebraic Geometry Am. Math. Soc. Transl. Ser. 2, vol. 202, pp. 131–149. American Mathematical Society (2001)Google Scholar
  12. 12.
    Kerckhoff S.: The asymptotic geometry of Teichmüller space. Topology 19, 23–41 (1980)MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Lenzhen A.: Teichmüller geodesics that do not have a limit in \({\mathcal{PMF}}\) . Geom. Topol. 12(1), 177–197 (2008)MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Liu, L., Su, W.: The horofunction compactification of Teichmüller metric, preprint,
  15. 15.
    Masur H.: On a class of geodesics in Teichmüller space. Ann. Math. 102, 205–221 (1975)MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Masur H.: Interval exchange transformations and measured foliations. Ann. Math. 115, 169–200 (1982)MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Masur H.: Two boundaries of Teichmüller space. Duke Math. 49, 183–190 (1982)MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Masur, H., Tabachinikov, S.: Rational Billiards and Flat structures, Handbook of dynamical systems, vol. 1A, p. 10151089 North-Holland, Amsterdam (2002)Google Scholar
  19. 19.
    McCarthy J., Papadopoulos A.: The visual sphere of Teichmüller space and a theorem of Masur-Wolf. Ann. Acad. Sci. Fenn. Math. 24, 147–154 (1999)MathSciNetGoogle Scholar
  20. 20.
    Minsky Y.: Extremal length estimates and product regions in Teichmüller space. Duke Math. 83, 249–286 (1996)MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    Miyachi H.: Teichmüller rays and the Gardiner–Masur boundary of Teichmüller space. Geom. Dedicata 137, 113–141 (2008)MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Miyachi, H.: Gardiner–Masur boundary of Teichmuller space: vanishing subsurfaces and Uniquely ergodic boundary points, preprint,
  23. 23.
    Penner R., Harer J.: Combinatorics of Train Tracks. Annals of Mathematics Studies, vol. 125. Princeton University Press, Princeton (1992)Google Scholar
  24. 24.
    Rees, M.(1982) An alternative approach to the ergodic theory of measured foliations on surfaces. Ergod. Theory Dyn. Syst. 1 (1981)(4), 461–488Google Scholar
  25. 25.
    Rieffel M.: Group C*-algebra as compact quantum metric spaces. Doc. Math. 7, 605–651 (2002)MathSciNetMATHGoogle Scholar
  26. 26.
    Strebel K.: Quadratic Differentials. Springer, Berlin (1984)MATHGoogle Scholar

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© Springer Science+Business Media B.V. 2012

Authors and Affiliations

  1. 1.Department of Mathematics, Graduate School of ScienceOsaka UniversityToyonaka, OsakaJapan

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