Skip to main content
SpringerLink
Log in

Cutting sequences, regular polygons, and the Veech group

  • Original Paper
  • Published:
Geometriae Dedicata Aims and scope Submit manuscript

Abstract

We describe the cutting sequences associated to geodesic flow on regular polygons, in terms of a combinatorial process called derivation. This work is an extension of some of the ideas and results in Smillie and Ulcigrai’s recent paper, where the analysis was made for the regular octagon. It turns out that the main structural properties of the octagon generalize in a natural way.

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. Arnoux, P.: Sturmian sequences. Substitutions in dynamics, arithmetics and combinatorics, 143198. Lecture Notes in Mathematics, vol. 1794. Springer, Berlin (2002)

  2. Davis D., Fuchs D., Tabachnikov S.: Periodic trajectories in the regular pentagon. Moscow Math J. 11, 1–23 (2011)

    MathSciNet  Google Scholar 

  3. Lothaire M.: Sturmian Words, Algebraic Combinatorics on Words, pp. 40–97. Cambridge University Press, Cambridge (2002)

    Google Scholar 

  4. Masur H., Tabachnikov S.: Rational Billiards and Flat Structures, Handbook of dynamical systems, vol. 1A, pp. 1015–1089. North-Holland, Amsterdam (2002)

    Google Scholar 

  5. Series C.: The geometry of Markoff numbers. Math. Intell. 7(3), 20–29 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  6. Smillie J., Ulcigrai C.: Beyond Sturmian sequences: coding linear trajectories in the regular octagon. Proc. Lond. Math. Soc. 102(2), 291–340 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  7. Smillie, J., Ulcigrai, C.: Geodesic flow on the Teichmueller disk of the regular octagon, cutting sequences and octagon continued fractions maps. In: Dynamical Numbers: Interplay between Dynamical Systems and Number Theory, Contemporary Mathematics, vol. 532, pp. 29–65. American Mathematical Society, Providence (2010)

  8. Smillie J., Weiss B.: Veech’s dichotomy and the lattice property. Ergodic Theory Dyn. Syst. 28, 1959–1972 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  9. Veech W.: Teichmüller curves in moduli space, Eisenstein series and an application to triangular billiards. Invent. Math. 87, 553–583 (1989)

    Article  MathSciNet  Google Scholar 

  10. Vorobets, Y.B.: Plane structures and billiards in rational polygons: the Veech alternative. Uspekhi Mat. Nauk 51(5), (311), 342 (1996); transl. Russian Math. Surv. 51(5), 779817 (1996)

  11. Zemlyakov A.N., Katok A.B.: Topological transitivity of billiards in polygons. Mat. Zametki 18(2), 291–300 (1975)

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Brown University, 151 Thayer Street, Providence, RI, 02912, USA

    Diana Davis

Authors
  1. Diana Davis
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Diana Davis.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Davis, D. Cutting sequences, regular polygons, and the Veech group. Geom Dedicata 162, 231–261 (2013). https://doi.org/10.1007/s10711-012-9724-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10711-012-9724-2

Keywords

Mathematics Subject Classification

Search

Navigation