On the Non-uniform Hyperbolicity of the Kontsevich-zorich Cocycle for Quadratic Differentials

We prove the non-uniform hyperbolicity of the Kontsevich-Zorich cocycle for a measure supported on abelian differentials which come from nonorientable quadratic differentials through a standard orienting, double cover construction. The proof uses Forni’s criterion [For11] for non-uniform hyperbolicity of the cocycle for SL(2,R)-invariant measures. We apply these results to the study of deviations in homology of typical leaves of the vertical and horizontal (non-orientable) foliations and deviations of ergodic averages. It is well known that the properties of a geodesic foliation (or flow) on a flat 2torus are completely characterized by its slope, whereas for a flat surface of higher genus the situation is far from similar. Such Riemann surface M of genus greater than one with a flat metric outside finitely many singularities can be given a pair of transverse, measured foliations (in the sense of Thurston). If such foliations are orientable, Zorich [Zor99] detected numerically that homology classes of segments of typical leaves of the foliation deviate from the asymptotic cycle (which is defined as the limit of normalized segments of leaves) in an unprecedented way, and that the rate of deviations are given by the positive Lyapunov exponents of the KontsevichZorich cocycle. Based on numerical experiments, the Kontsevich-Zorich conjecture was formulated, which claimed that for Lebesgue-almost all classes of conformally equivalent flat metrics with orientable foliations, the exponents are all distinct and non-zero. In other words, the cocycle is non-uniformly hyperbolic and has a simple spectrum. It was also conjectured that there should be similar deviation phenomena for ergodic averages of functions in some space of functions. The first proof of the non-uniform hyperbolicity of the Kontsevich-Zorich cocycle came from Forni [For02], but the simplicity question remained open for surfaces of genus greater than 2. The full conjecture was finally proved through methods completely different from those of Forni by Avila and Viana [AV07]. In [For02], a complete picture is painted on the deviations of ergodic averages along the straight line flows given by vector fields tangent to the foliations on the flat surface. The rate of divergence of such deviations are also described by all of the Lyapunov exponents of the Kontsevich-Zorich cocycle. In this paper we study the same phenomena for the case of non-orientable foliations on flat surfaces. Although there is no vector field to speak of, we can still describe deviations of integrals of functions along leaves of the foliation. Our work has been made substantially easier by the recent criterion of Forni [For11], where the proof of non-uniform hyperbolicity in [For02] has been condensed and Date: May 16, 2012. Supported by the Brin and Flagship Fellowships at the University of Maryland.