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arXiv · 1009.4655

A geometric criterion for the non-uniform hyperbolicity of the Kontsevich--Zorich cocycle

Giovanni Forni
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Abstract

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 Kontsevich--Zorich cocycle on the real Hodge bundle. Applications include measures supported on the $\SL$-orbits of all algebraically primitive Veech surfaces (see also \cite{Bouw:Moeller}) and of all Prym eigenforms discovered in \cite{McMullen2}, as well as all canonical absolutely continuous measures on connected components of strata of the moduli space of abelian differentials (see also \cite{Ftwo}, \cite{Avila:Viana}). The argument simplifies and generalizes our proof for the case of canonical measures \cite{Ftwo}. In an Appendix Carlos Matheus discusses several relevant examples which further illustrate the power and the limitations of our criterion.

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Giovanni Forni. 2010-09-23. A geometric criterion for the non-uniform hyperbolicity of the Kontsevich--Zorich cocycle. https://arxiv.org/abs/1009.4655

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