Chapter 8 On the lyapunov exponents of the Kontsevich-Zorich cocycle

Abstract

The Kontsevich-Zorich cocycle is a cocycle over the Teichmtiller flow on the moduli space of holomorphic (quadratic) differentials. The study of the dynamics of this cocycle, particularly, of its Lyapunov structure, has important applications to the ergodic theory of interval exchange transformations (i.e.t.'s) and related systems, such as measured foliations, flows on translation surfaces, and rational polygonal billiards. The Kontsevich-Zorich cocycle is a continuoustime version of a cocycle introduced by G. Rauzy as a “continued fractions algorithm.” Zorich made the key discovery that typical trajectories of generic (orientable) measured foliations on the surfaces of higher genus (or equivalently of genetic i.e.t.'s with at least 4 intervals) deviate from the mean according to a power law with exponents determined by the Lyapunov exponents of the cocycle. He also observed that, as a consequence of the close connection between the cocycle and the Teichmtiller geodesic flow, the simplicity of the top exponent, sometimes called the spectral gap property, is equivalent to the (nonuniform) hyperbolicity of the Teichmtiller flow. The role of the Kontsevich-Zorich cocycle can be explained by the somewhat vague observation that it provides a renormalization dynamics for i.e.t.'s (and related systems). Such systems provide fundamental examples of parabolic dynamics that by definition is characterized by sub-exponential (polynomial) divergence of nearby orbits.