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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.

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Correspondence to Rodrigo Treviño.

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Supported by the Brin and Flagship Fellowships at the University of Maryland.

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Treviño, R. On the non-uniform hyperbolicity of the Kontsevich–Zorich cocycle for quadratic differentials. Geom Dedicata 163, 311–338 (2013). https://doi.org/10.1007/s10711-012-9751-z

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  • DOI: https://doi.org/10.1007/s10711-012-9751-z

Keywords

  • Quadratic differentials
  • Kontsevich–Zorich cocycle
  • Non-orientable foliations

Mathematics Subject Classification

  • 32G15
  • 37D25