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Lagrange spectra for translation surfaces via renormalization
Tue 05 March 2013, 14:00
Luca Marchese
Paris 13
Organiser: Jens Marklof
ABSTRACT
ABSTRACT: We introduce Lagrange spectra for closed affine-invariant loci of 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 generally 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 invariant loci of translation surfaces. Moreover we prove two explicit formulas 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. This is joint work with Pascal Hubert and Corinna Ulcigrai.