Geometriae Dedicata

, Volume 190, Issue 1, pp 53–80 | Cite as

Cutting sequences on square-tiled surfaces

Original Paper

Abstract

We characterize cutting sequences of infinite geodesics on square-tiled surfaces by considering interval exchanges on specially chosen intervals on the surface. These interval exchanges can be thought of as skew products over a rotation, and we convert cutting sequences to symbolic trajectories of these interval exchanges to show that special types of combinatorial lifts of Sturmian sequences completely describe all cutting sequences on a square-tiled surface. Our results extend the list of families of surfaces where cutting sequences are understood to a dense subset of the moduli space of all translation surfaces.

Keywords

Cutting sequences Dynamical systems Square-tiled surfaces Translation surfaces 

Mathematics Subject Classification (1991)

Primary 37E35 

References

  1. 1.
    Arnoux, P.: Sturmian sequences. In: Berthé, V,. Ferenczi, S., Mauduit, C., Siegel, A. (eds.) Substitutions in Dynamics, Arithmetics and Combinatorics, volume 1794 of Lecture Notes in Mathematics, chapter 6, pp. 143–198. Springer-Verlag, Berlin (2002)Google Scholar
  2. 2.
    Davis, D.: Cutting sequences, regular polygons, and the Veech group. Geom. Dedicata 162, 231–261 (2013)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Davis, D.: Cutting sequences on translation surfaces. New York J. Math. 20, 399–429 (2014)MathSciNetMATHGoogle Scholar
  4. 4.
    Davis, D., Pasquinelli, I., Ulcigrai, C.: Cutting sequences on Bouw-Möller surfaces: an S-adic characterization. ArXiv e-prints, (September 2015), arXiv:1509.03905
  5. 5.
    Ferenczi, S., Zamboni, L.Q.: Languages of \(k\)-interval exchange transformations. Bull. Lond. Math. Soc. 40(4), 705–714 (2008)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Gutkin, E., Judge, C.: Affine mappings of translation surfaces: geometry and arithmetic. Duke Math. J. 103(2), 191–213 (2000)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Keane, M.: Interval exchange transformations. Math. Z. 141, 25–31 (1975)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Kerckhoff, S., Masur, H., Smillie, J.: Ergodicity of billiard flows and quadratic differentials. Ann. of Math. 124(2), 293–311 (1986)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Morse, M., Hedlund, G.A.: Symbolic dynamics. Am. J. Math. 60(4), 815–866 (1938)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Masur, H., Tabachnikov, S.: Rational billiards and flat structures. In: Handbook of Dynamical Systems, Vol. 1A, pp. 1015–1089. North-Holland, Amsterdam (2002)Google Scholar
  11. 11.
    Schmithüsen, G.: An algorithm for finding the Veech group of an origami. Exp. Math. 13(4), 459–472 (2004)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Series, C.: The geometry of Markoff numbers. Math. Intell. 7(3), 20–29 (1985)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Smillie, J., Ulcigrai, C.: Beyond Sturmian sequences: coding linear trajectories in the regular octagon. Proc. Lond. Math. Soc. (3) 102(2), 291–340 (2011)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Veech, W.A.: Teichmüller curves in moduli space, Eisenstein series and an application to triangular billiards. Invent. Math. 97(3), 553–583 (1989)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Wright, A.: Translation surfaces and their orbit closures: An introduction for a broad audience. EMS Surv. Math. Sci. 2(1), 63–108 (2015)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Wu, S.J., Zhong, Y.M.: On cutting sequences of the \(L\)-shaped translation surface tiled by three squares. Sci. China Math. 58(6), 1311–1326 (2015)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Zorich, A.: Flat Surfaces. Frontiers in Number Theory, Physics, and Geometry. I. Springer, Berlin, pp. 437–583 (2006)Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2017

Authors and Affiliations

  1. 1.Wake Forest UniversityWinston-SalemUSA

Personalised recommendations