Hamid Al-Saqban: On the Unique Ergodicity for Foliations of Generic Abelian Differentials

Abstract: In Teichmuller theory, a theorem due to H. Masur and W. Veech states that for generic Abelian differentials, the leaves of the horizontal foliation are uniquely ergodic. Said differently, the orbits of the horizontal straight-line flow on a generic translation surface are uniquely ergodic. Subsequently, G. Forni proved an effective form of this statement, establishing in particular precise power-laws for the deviations of ergodic averages of smooth functions from the power law (as predicted by the ergodic theorem). The main goal of this talk is to introduce and motivate translation surfaces, and to explain the main ideas behind a (microlocal) analytic approach to effective unique ergodicity for straight-line flows on a generic translation surface. This is a joint work in progress with D. Galli (University of Zurich).