Sala Conferenze (Puteano, Centro De Giorgi).
We consider two dynamical problems for translation surfaces, both related to diophantine approximations: one is the study of the asymptotic amplitude of excursions at infinity of a Teichmuller geodesic in parameter space, the other is the study of recurrence for a linear flow, like for example the billiard flow in a rational polygon. In an abstract setting, we present metric criteria leading to a generalized version of Khinchin-Jarnick theorem and Jarnick’s inequality, where the former is a dichotomy for the Hausdorff measure of well approximable numbers and the latter gives upper and lower bounds for the dimension of badly approximable numbers. Finally we prove that the abstract metric criteria are satisfied on any given translation surface. This is a joint work with Rodrigo Trevino and Steffen Weil, based on previuos works of Beresnevich-Velani and Minsky-Weiss.