Skip to main content
SpringerLink
Log in

Parking garages with optimal dynamics

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

Abstract

We construct generalized polygons (‘parking garages’) in which the billiard flow satisfies the Veech dichotomy, although the associated translation surface obtained from the Zemlyakov–Katok unfolding is not a lattice surface. We also explain the difficulties in constructing a genuine polygon with these properties.

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., Schmidt T.: Veech surfaces with non-periodic directions in the trace field. J. Mod. Dyn. 3(4), 611–629 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bouw I., Möller M.: Teichmüller curves, triangle groups, and Lyapunov exponents. Ann. Math. 172, 139–185 (2010)

    Article  MATH  Google Scholar 

  3. Cheung Y., Hubert P., Masur H.: Topological dichotomy and strict ergodicity for translation surfaces. Ergod. Theory Dyn. Syst. 28, 1729–1748 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  4. Cohen, M.: Looking for a Billiard Table which is not a Lattice Polygon but satisfies the Veech dichotomy, M. Sc. thesis, Ben-Gurion University (2010). \({\tt arxiv.org/pdf/1011.3217}\)

  5. DeMarco L.: The conformal geometry of billiards. Bull. AMS 48(1), 33–52 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  6. Eskin A., Marklof J., Morris D.: Unipotent flows on the space of branched covers of Veech surfaces. Ergod. Theorm Dyn. Syst. 26(1), 129–162 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  7. Fox R.H., Kershner R.B.: Concerning the transitive properties of geodesics on a rational polyhedron. Duke Math. J. 2(1), 147–150 (1936)

    Article  MathSciNet  Google Scholar 

  8. Gutkin E., Hubert P., Schmidt T.: Affine diffeomorphisms of translation surfaces: Periodic points, Fuchsian groups, and arithmeticity. Ann. Sci. École Norm. Sup. (4) 36, 847–866 (2003)

    MathSciNet  MATH  Google Scholar 

  9. Hubert P., Schmidt T.: Infinitely generated Veech groups. Duke Math. J. 123(1), 49–69 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  10. Masur, H., Tabachnikov, S.: Rational billiards and flat structures. In: Handbook of dynamical systems, vol. 1A, pp. 1015–1089. North-Holland, Amsterdam (2002)

  11. Smillie J., Weiss B.: Veech dichotomy and the lattice property. Ergod. Theorm. Dyn. Syst. 28, 1959–1972 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  12. Veech W.A.: Teichmüller curves in moduli space, Eisenstein series and an application to triangular billiards. Invent. Math. 97, 553–583 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  13. Vorobets Y.: Planar structures and billiards in rational polygons: the Veech alternative. (Russian); translation in Russian Math. Surveys 51(5), 779–817 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  14. Ward C.C.: Calculation of Fuchsian groups associated to billiards in a rational triangle. Ergod. Theory Dyn. Syst. 18, 1019–1042 (1998)

    Article  MATH  Google Scholar 

  15. Zemlyakov, A., Katok, A.: Topological transitivity of billiards in polygons, Math. Notes USSR Acad. Sci: 18:2 291–300 (1975). (English translation in Math. Notes 18:2 760–764)

  16. Zorich, A. : Flat surfaces. In: Cartier, P., Julia, B., Moussa, P., Vanhove, P. (eds.) Frontiers in number theory, physics and geometry, Springer, Berlin (2006)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Ben Gurion University, 84105, Be’er Sheva, Israel

    Meital Cohen & Barak Weiss

Authors
  1. Meital Cohen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Barak Weiss
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Barak Weiss.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Cohen, M., Weiss, B. Parking garages with optimal dynamics. Geom Dedicata 161, 157–167 (2012). https://doi.org/10.1007/s10711-012-9699-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10711-012-9699-z

Keywords

Mathematics Subject Classification (2000)

Search

Navigation