The Lagrange spectrum of some square-tiled surfaces

  • Pascal Hubert
  • Samuel Lelièvre
  • Luca Marchese
  • Corinna Ulcigrai


Lagrange spectra have been defined for closed submanifolds of the moduli space of translation surfaces which are invariant under the action of SL(2, R). We consider the closed orbit generated by a specific covering of degree 7 of the standard torus, which is an element of the stratum H(2). We give an explicit formula for the values in the spectrum, in terms of a cocycle over the classical continued fraction. Differently from the classical case of the modular surface, where the lowest part of the Lagrange spectrum is discrete, we find an isolated minimum, and a set with a rich structure right above it.


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  1. [AMU]
    M. Artigiani, L. Marchese and C. Ulcigrai, Persistent Hall rays for Lagrange spectra at cusps of Riemann surfaces, arXiv:1710.02042.Google Scholar
  2. [AMU16]
    M. Artigiani, L. Marchese and C. Ulcigrai, The Lagrange spectrum of a Veech surface has a Hall ray, Groups, Geometry, and Dynamics 10 (2016), 1287-1337.Google Scholar
  3. [BD17]
    M. Boshernitzan and V. Delecroix, From a packing problem to quantitative recurrence in [0, 1] and the Lagrange spectrum of interval exchanges, Discrete Analysis (2017), Paper No. 10, 25.Google Scholar
  4. [CF89]
    T. W. Cusick and M. E. Flahive, The Markoff and Lagrange Spectra, Mathematical Surveys and Monographs, Vol. 30, American Mathematical Society, Providence, RI, 1989.Google Scholar
  5. [CMM]
    A. Cerqueira, C. Matheus and C. G. Moreira, Continuity of hausdorff dimension across dynamical lagrange and markov spectra, arXiv:1602:04649.Google Scholar
  6. [FM14]
    G. Forni and C. Matheus, introduction to Teichmüller theory and its applications to dynamics of interval exchange transformations, Bows on surfaces and billiards, Journal of Modern Dynamics 8 (2014), 271-436.Google Scholar
  7. [Hal47]
    M. Hall, Jr., On the sum and product of continued fractions, Annals of Mathematics 48 (1947), 966-993.Google Scholar
  8. [HL06]
    P. Hubert and S. Lelièvre, Prime arithmetic Teichmüller discs in H(2), Israel Journal of Mathematics 151 (2006), 281-321.Google Scholar
  9. [HMU15]
    P. Hubert, L. Marchese and C. Ulcigrai, Lagrange spectra in Teichmüller dynamics via renormalization, Geometric and Functional Analysis 25 (2015), 180-255.Google Scholar
  10. [McM05]
    C. T. McMullen, Teichmüller curves in genus two: discriminant and spin, Mathematische Annalen 333 (2005), 87-130.Google Scholar
  11. [Mor]
    C. G. Moreira, Geometric properties of Markov and Lagrange spectra, preprint, IMPA.Google Scholar
  12. [PP10]
    J. Parkkonen and F. Paulin, Prescribing the behaviour of geodesics in negative curvature, Geometry & Topology 14 (2010), 277-392.Google Scholar
  13. [RM17]
    S. Romaña and C. G. Moreira, On the Lagrange and Markov dynamical spectra, Ergodic Theory and Dynamical Systems 37 (2017), 1570-1591.Google Scholar
  14. [Ser85]
    C. Series, The geometry' of Markoff numbers, Mathematical Intelligencer 7 (1985), 20-29.Google Scholar
  15. [SW07]
    J. Smillie and B. Weiss, Finiteness results for flat surfaces: a survey and problem list, in Partially Hyperbolic Dynamics, Laminations, and Teichmüller Flow, Fields Institute Communications, Vol. 51, American Mathematical Society, Providence, RI, 2007, pp. 125-137.Google Scholar
  16. [Vee95]
    W. A. Veech, Geometric realizations of hvperelliptic curves, in Algorithms, fractals, and dynamics (Okavama/Kvoto, 1992), Plenum, New York, 1995, pp. 217-226.Google Scholar
  17. [Zor06]
    A. Zorich, Flat surfaces, in Frontiers in Number Theory, Physics, and Geometry. 1, Springer, Berlin, 2006, pp. 437-583.Google Scholar

Copyright information

© Hebrew University of Jerusalem 2018

Authors and Affiliations

  • Pascal Hubert
    • 1
  • Samuel Lelièvre
    • 2
  • Luca Marchese
    • 3
  • Corinna Ulcigrai
    • 4
  1. 1.I2M, Centre de Mathématiques et Informatique (CMI)Université Aix-MarseilleMarseille Cedex 13France
  2. 2.Laboratoire de mathématique d’Orsay, UMR 8628 CNRS Université Paris-SudOrsay cedexFrance
  3. 3.Université Paris 13, Sorbonne Paris Cité, LAG A, UMR 7539VilletaneuseFrance
  4. 4.School of MathematicsUniversity of Bristol, University WalkClifton, BristolUK

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