### Hall rays for Lagrange spectra of Veech surfaces

**Ulcigrai, C ***(University of Bristol)*

Monday 23 June 2014, 13:30-14:30

Seminar Room 2, Newton Institute Gatehouse

#### Abstract

The Lagrange spectrum is a subset of the real line which can be described in terms of badly approximable numbers. In joint work with Hubert and Marchese, we defined a generalization of Lagrange spectra for translation surfaces and badly approximable interval exchange transformations. Both in the classical case and in the case of Veech translation surfaces, Lagrange spectra can be also described in terms of asymptotic depths penetration of hyperbolic geodesics in the cusps of the associated hyperbolic surface (the Teichmueller curve). In joint work with Mauro Artigiani and Luca Marchese, we show that high Lagrange spectra of Veech surfaces can be calculated using a symbolic coding in the sense of Bowen and Series. We then prove that the Lagrange spectrum of any Veech translation surface contains a semi-line, i.e. a Hall's ray.

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