Beyond orientability and compactness: new results on the dynamics of flat surfaces

In the first part, 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. The proof uses Forni's criterion 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. In the second part, we prove an ergodic theorem for flat surfaces of finite area whose Teichmuller orbits are recurrent to a compact set of SL(2;R)/SL(S), where SL(S) is the Veech group of the surface. In this setting, this means that the translation flow on a flat surface can be renormalized through its Veech group. This result applies in particular to flat surfaces of infinite genus and finite area, and we apply our result to existing surfaces in the literature to prove that the corresponding foliations of the surface corresponding to a periodic or recurrent Teichmuller orbit are ergodic