Abstract:
The Lagrange spectrum is a classical object in Diophantine approximation on the real line that has been generalised to many different settings. In particular, recently it has been generalised to a similar object for translation surfaces, which attracted quite some attention in the field. We study the Lagrange spectrum in the contest of Veech translation surfaces. These are particular translation surfaces with many symmetries, that can be thought as a dynamical equivalent of the torus in higher genera.
Together with L. Marchese and C. Ulcigrai, we show that for such surfaces, similarly to what happens in the classical case, the Lagrange spectrum contains an infinite interval, called Hall ray. In our construction we use coding of hyperbolic geodesics and we deduce a formula that allows to describe high values in the spectrum as a sum of two Cantor sets.
Time permitting, we will also talk about an ongoing project about a specific class of examples of Veech surfaces: the surfaces obtained by gluing a regular 2n-gon. In this case one can prove finer result, for instance studying the minimum of the spectrum, the so-called Hurwitz constant.