On the Non-Uniform Hyperbolicity of the Kontsevich-Zorich Cocycle for Quadratic Differentials

TL;DR

This paper proves the non-uniform hyperbolicity of the Kontsevich-Zorich cocycle for certain quadratic differentials, extending understanding of dynamical stability and deviations in related foliations.

Contribution

It establishes non-uniform hyperbolicity for the cocycle in non-orientable quadratic differential cases using Forni's criterion.

Findings

Non-uniform hyperbolicity for measures from non-orientable quadratic differentials

Applications to deviations in homology of foliations

Insights into ergodic averages deviations

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 orientating, double cover construction. 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.