Consider a component ${\cal Q}$ of a stratum in the moduli space of area-one abelian differentials on a surface of genus g. Call a property ${\cal P}$ for periodic orbits of the Teichmüller flow on ${\cal Q}$ typical if the growth rate of orbits with property ${\cal P}$ is maximal. We show that the following property is typical. Given a continuous integrable cocycle over the Teichmüller flow with values in a vector bundle $V\to {\cal Q}$ , the logarithms of the eigenvalues of the matrix defined by the cocycle and the orbit are arbitrarily close to the Lyapunov exponents of the cocycle for the Masur–Veech measure.

In this survey we prove the sharpest results on the loss of Sobolev regularity for solutions of the cohomological equation for translation flows on translation surfaces, available to the methods developed by the author in Forni [Solutions of the cohomological equation for area-preserving flows on compact surfaces of higher genus. Ann. of Math. (2)146(2) (1997), 295–344] and Forni [Deviation of ergodic averages for area-preserving flows on surfaces of higher genus. Ann. of Math. (2)155(1) (2002), 1–103]. The paper was mostly written between 2005 and 2006 while the author was at the University of Toronto, Canada, and was posted on arXiv in July 2007 [Forni. Sobolev regularity of solutions of the cohomological equation. Preprint, 2007, arXiv:0707.0940v2]. In an updated introduction we describe our results, taking into account later work on the problem and relevant recent progress in the field of Teichmüller dynamics, interval exchange transformations and translation flows.