Introduction 1. The Kontsevich-Zorich cocycle 2. Variational formulas 3. A lower bound for the second Lyapunov exponent 4. The determinant locus 5. The Kontsevich-Zorich formula revisited and other… (More)

There are infinitely many obstructions to the existence of smooth solutions of the cohomological equation Uu = f , where U is the vector field generating the horocycle flow on the unit tangent bundle… (More)

We study the equation Xu = f where X belongs to a class of area-preserving vector elds, having saddle-type singularities, on a compact orientable surface M of genus g 2. For a \full measure" set of… (More)

We show that a typical interval exchange transformation is either weakly mixing or it is an irrational rotation. We also conclude that a typical translation flow on a surface of genus g ≥ 2 (with… (More)

Let S be a closed topological surface of genus g and let Ω denote (a connected component of) a stratum of the moduli space of abelian differentials on S. We prove a polynomial upper bound on the… (More)

We construct an orientable holomorphic quadratic differential on a Riemann surface of genus 4 whose SL(2, R)-orbit is closed and has a highly degenerate Kontsevich-Zorich spectrum. This example is… (More)

The main results of this paper are limit theorems for horocycle flows on compact surfaces of constant negative curvature. One of the main objects of the paper is a special family of… (More)

Let X be a vector field on a compact connected manifold M . An important question in dynamical systems is to know when a function g : M → R is a coboundary for the flow generated by X , i.e. when… (More)

that is, the problem of finding a function u on M whose derivative along the flow is equal to a given function f on M . Roughly speaking, the Ccohomology of the flow is the space of non-trivial… (More)

We prove a polynomial upper bound on the deviation of ergodic averages for almost all directional flows on every translation surface, in particular, for the generic directional flow of billiards in… (More)