Quantitative density statements for translation surfaces

Abstract

The main results in this thesis are quantitative descriptions of the orbits of two dynamical systems on translation surfaces. First, we study the action of a discrete subgroup of $SL_2(\R)$ on a closed square-tiled surface and quantify the density of the orbits by proving a Diophantine estimate. Second, we study the linear flow on a translation surface and identify a quantitative density condition on the flow that is equivalent to the boundedness of an associated geodesic in the moduli space of translation surfaces.