CRM: Centro De Giorgi
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Continued Fractions, Interval Exchanges and Applications to Geometry

Lagrange spectra for translation surfaces

speaker: Luca Marchese (Université Paris 13)

abstract: We introduce Lagrange spectra for translation surfaces, generalizing the classical Lagrange spectrum. Several generalizations of the classical Lagrange spectrum exist in the literature, defined in terms of some flow in parameter space and satisfying some basic properties: the existence of an Hall's ray, the closedness of the spectrum and the density of values coming from periodic orbits. We focus on these properties for the Teichmuller flow on closed an SL(2,R)-invariant subsets of the moduli space of translation surfaces. We prove two explicit formulae that allow to compute the spectra in terms of two continued fraction algorithms: the first is a skew-product over the classical continued fraction, which applies to arithmetic Teichmuller discs, the second is the so-called Rauzy-Veech induction, which applies to strata of translation surfaces, and, up to some extent, also to any affine invariant locus. This is joint work with Pascal Hubert and Corinna Ulcigrai.

Fri 14 Jun, 9:10 - 10:00, Aula Dini
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