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

By G. Forni. 2010

51 citations recorded by the source corpus.

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.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/fa27bc7be8b2ff46f6287598d78eea7fbd5f1590

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Citations in the corpus

Sum of Lyapunov exponents of the Hodge bundle with respect to the Teichmüller geodesic flow (2011, background): “tsrelated to the study ofthe Lyapunovspectrum remain beyond the scope of our consideration. We address the reader to the original paper of G. Forni [Fo1], to the survey [Fo2] and to the recent papers [F3], [T], [Au1], [Au2] for the very important issues of determinant locus and of nonuniform hyperbo”
Introduction to Teichm\"uller theory and its applications to dynamics of interval exchange transformations, flows on surfaces and billiards (2013, methodology): “Forni extended some techniques of his article [29] to prove in [31] a geometric criterion for the non-uniform hyperbolicity of the KZ cocycle (i.”
Limit theorems for translation flows (2008, unknown): “Forni [22], and C.”
Lyapunov spectrum of square-tiled cyclic covers (2010, background): “XESKIN,MAXIMKONTSEVICH,ANDANTONZORICH Appendix C. Homological dimension of the square-tiled MN(N− 1,1,N− 1,1) The following characteristic of a square-tiled surface is important for applications, see [F3]. Given a square-tiled surface S0, consider a collection of cycles c1,...,cj representing the wa”
A criterion for the simplicity of the Lyapunov spectrum of square-tiled surfaces (2013, methodology): “By Forni’s criterion [22], it follows that all Lyapunov exponents of any origami O of the form indicated in Fig.”