Lagrange spectra for translation surfaces

The classical Lagrange spectrum (and the closely related Markoff spectrum), is a well studied subset of the reals which can be defined either arithmetically, in terms of Diophantine approximation for badly approximable numbers, or geometrically, considering the asymptotic penetration of bounded geodesics on the modular surface. We will consider a generalization of the Lagrange spectrum to the context of translation surfaces and interval exchange transformations (which generalize respectively flat tori and rotations). In the special case of Veech translation surfaces, as in the classical case, Lagrange spectra has also an interpretation in terms of asymptotic depths penetration of hyperbolic geodesics in the cusps of the associated hyperbolic surface (the Teichmueller curve). To prove some properties of Lagrange spectra of translation surfaces by exploiting a multi-dimensional continued fraction algorithm (Rauzy-Veech induction). In the special case of Veech translation surfaces, we show that high Lagrange spectra of Veech surfaces can be calculated using a symbolic coding in the sense of Bowen and Series. In particular, we prove that the Lagrange spectrum of any invariant locus of translation surfaces contains a semi-line, i.e. a Hall's ray. The talk is based on joint works with P. Hubert and L. Marchese and M. Artigiani and L. Marchese.