Dynamics in the Moduli Space of Abelian Differentials

We announce the proof of the Zorich–Kontsevich conjecture: the nontrivial Lyapunov exponents of the Teichmüller flow on (any connected component of a stratum of) the moduli space of Abelian differentials on compact Riemann surfaces are all distinct. By previous work of those authors, this implies the existence of the complete asymptotic Lagrangian flag describing the behavior in homology of the vertical foliation in a typical translation surface. 1 – Abelian differentials 1.1. An Abelian differential on a compact Riemann surface M is a holomorphic complex 1-form ω on the surface. In local coordinates z, it may be written ωz = φ(z) dz where the coefficient φ is a holomorphic function. Given another local coordinate w, the corresponding local expression ωw = ψ(w)dw is determined by ψ(w) = φ(z) dz dw on the intersection of the coordinate domains. 1.2. We assume that the Abelian differential is not identically zero. Then its zeros are isolated and, hence, finitely many. Let them be z1, ..., zκ, with κ ≥ 0. Received : July 18, 2005. 532 ARTUR AVILA and MARCELO VIANA Near any point p such that ωp is non-zero, we may always find adapted local coordinates (1) ζ = ∫ z