On the non-uniform hyperbolicity of the Kontsevich–Zorich cocycle for quadratic differentials

Abstract

We prove the non-uniform hyperbolicity of the Kontsevich–Zorich cocycle for a measure supported on abelian differentials which come from non-orientable quadratic differentials through a standard orienting, double cover construction. The proof uses Forni’s criterion (J. Mod. Dyn. 5(2):355–395, 2011) for non-uniform hyperbolicity of the cocycle for \({SL(2, \mathbb{R})}\)-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.