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

@article{Trevio2010OnTN,
  title={On the non-uniform hyperbolicity of the Kontsevich–Zorich cocycle for quadratic differentials},
  author={Rodrigo Trevi{\~n}o},
  journal={Geometriae Dedicata},
  year={2010},
  volume={163},
  pages={311-338}
}
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… 
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References

SHOWING 1-10 OF 30 REFERENCES
A geometric criterion for the non-uniform hyperbolicity of the Kontsevich--Zorich cocycle
We prove a geometric criterion on a $\SL$-invariant ergodic probability measure on the moduli space of holomorphic abelian differentials on Riemann surfaces for the non-uniform hyperbolicity of the
Simplicity of Lyapunov spectra: proof of the Zorich-Kontsevich conjecture
We prove the Zorich–Kontsevich conjecture that the non-trivial Lyapunov exponents of the Teichmüller ow on (any connected component of a stratum of) the moduli space of Abelian differentials on
Parity of the spin structure defined by a quadratic differential
According to the work of Kontsevich{Zorich, the invariant that classies nonhyperelliptic connected components of the moduli spaces of Abelian dierentials with prescribed singularities, is the parity
Deviation of ergodic averages for area-preserving flows on surfaces of higher genus
We prove a substantial part of a conjecture of Kontsevich and Zorich on the Lyapunov exponents of the Teichmuller geodesic flow on the deviation of ergodic averages for generic conservative flows on
Exponential Mixing for the Teichmuller flow in the Space of Quadratic Differentials
We consider the Teichmuller flow on the unit cotangent bundle of the moduli space of compact Riemann surfaces with punctures. We show that it is exponentially mixing for the Ratner class of
Dynamics and geometry of the Rauzy–Veech induction for quadratic differentials
Abstract Interval exchange maps are related to geodesic flows on translation surfaces; they correspond to the first return maps of the vertical flow on a transverse segment. The Rauzy–Veech induction
Quadratic differentials with prescribed singularities and pseudo-Anosov diffeomorphisms
A quadratic differential on a Riemann surface M determines certain " topological" data: the genus o f M; the orders o f zeros and poles; and the orientability o f the horizontal foliation. In this
Euler characteristics of Teichmüller curves in genus two
We calculate the Euler characteristics of all of the Teichmuller curves in the moduli space of genus two Riemann surfaces which are generated by holomorphic one-forms with a single double zero. These
Connected components of the moduli spaces of Abelian differentials with prescribed singularities
Consider the moduli space of pairs (C,ω) where C is a smooth compact complex curve of a given genus and ω is a holomorphic 1-form on C with a given list of multiplicities of zeroes. We describe
Self-Inverses in Rauzy Classes
Thanks to works by M. Kontsevich and A. Zorich followed by C. Boissy, we have a classification of all Rauzy Classes of any given genus. It follows from these works that Rauzy Classes are closed under
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