Athreya, Jayadev SShokrieh, FarbodSoutherland, Joshua2022-07-142022-07-142022-07-142022Southerland_washington_0250E_24417.pdfhttp://hdl.handle.net/1773/49076Thesis (Ph.D.)--University of Washington, 2022The 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.application/pdfen-USCC BYDynamical systemsErgodic theoryGeometryRiemann surfacesTeichmuller dynamicsTranslation surfacesMathematicsMathematicsThesis