|Dept.||Univ. Paris 13|
The classical theory of diophantine approximations has two dynamical interpretations, the first in terms of recurrence for linear flows on flat tori, the second in terms of cusp excursions for the geodesic flow on the modular surface. These classical issues generalize to nice higher genus flat objects, called translation surfaces, as a part of the very active area of Teichmuller's dynamics. In an abstract setting, we prove a version of Khinchin-Jarnick Theorem, which gives a dichotomy for the Hausdorff measure of the set of points admitting very good diophantine approximations. In the same setting we give an estimate of the Hausdorff dimension of bad diophantine approximations. Finally we show that translation surfaces fit in our abstract setting. This is a joint work with Rodrigo Trevino and Steffen Weil.