Some results that hold on every flat surface

Recently Eskin-Mirzakhani-Mohammadi have proven a number of
powerful results about the $SL_2(R)$ orbits of abelian differentials and the $SL_2(R)$ ergodic measures on the stratum. We discuss some results motivated and enabled by this work. One result is that are that for every abelian differential there is a measure on the stratum, such
that after rotating in almost every direction, the geodesic flow equidistributes for this measure on the stratum. Another result is a criterion to understand the deviation of ergodic averages for the linear flow in almost every direction. This is verified for genus $2$
surfaces and thus has an application, being explored by others, to the windtree model. This is joint work with Alex Eskin.