The Lagrange spectrum of some square-tiled surfaces
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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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- [AMU]M. Artigiani, L. Marchese and C. Ulcigrai, Persistent Hall rays for Lagrange spectra at cusps of Riemann surfaces, arXiv:1710.02042.Google Scholar
- [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
- [CF89]T. W. Cusick and M. E. Flahive, The Markoff and Lagrange Spectra, Mathematical Surveys and Monographs, Vol. 30, American Mathematical Society, Providence, Rl, 1989.Google Scholar
- [CMM]A. Cerqueira, C. Matheus and C. G. Moreira, Continuity of hausdorff dimension across dynamical lagrange and markov spectra, arXiv:1602:04649.Google Scholar
- [Mor]C. G. Moreira, Geometric properties of Markov and Lagrange spectra, preprint, IMPA.Google Scholar
- [SW07]J. Smillie and B. Weiss, Finiteness results for flat surfaces: a survey and problem list, in Partially Hyperbolic Dynamics, Laminations, and Teichmiiller Flow, Fields Institute Communications, Vol. 51, American Mathematical Society, Providence, Rl, 2007, pp. 125–137.Google Scholar
- [Zor06]A. Zorich, Flat surfaces, in Frontiers in Number Theory, Physics, and Geometry. I, Springer, Berlin, 2006, pp. 437–583.Google Scholar