Resonant sets, diophantine conditions, and Hausdorff measures for translation surfaces

Luca Marchese, Rodrigo Treviño, Steffen Weil

Published 2015-02-17Version 1

We consider Teichm\"uller geodesics in strata of translation surfaces. We prove a Jarn\'ik-type inequality for geodesics bounded in some compact part of the stratum and we establish generalized logarithmic laws for geodesics admitting excursions to infinity at a given prescribed rate. Our main tool are planar resonant sets arising from a given translation surface, that is the countable sets of directions of its saddle connections or of its closed geodesics, filtered according to length. We study approximations of a general direction by the elements of a resonant set. In an abstract setting, and assuming specific metric properties for resonant sets, we prove a dichotomy for the Hausdorff measure of well approximable directions and an estimate on the dimension of badly approximable directions. Then we prove that resonant sets arising from a translation surface satisfies the required metric properties. Our techniques also give estimates on the Hausdorff dimension of the set of directions in a rational billiard having fast recurrence.