Journal of Modern Dynamics (JMD)

A geometric criterion for the nonuniform hyperbolicity of the Kontsevich--Zorich cocycle

Pages: 355 - 395, Issue 2, April 2011      doi:10.3934/jmd.2011.5.355

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Giovanni Forni - Department of Mathematics, University of Maryland, College Park, MD 20742-4015, United States (email)

Abstract: We establish a geometric criterion on a $SL(2, R)$-invariant ergodic probability measure on the moduli space of holomorphic abelian differentials on Riemann surfaces for the nonuniform hyperbolicity of the Kontsevich--Zorich cocycle on the real Hodge bundle. Applications include measures supported on the $SL(2, R)$-orbits of all algebraically primitive Veech surfaces (see also [7]) and of all Prym eigenforms discovered in [34], as well as all canonical absolutely continuous measures on connected components of strata of the moduli space of abelian differentials (see also [4, 17]). The argument simplifies and generalizes our proof for the case of canonical measures [17]. In the Appendix, Carlos Matheus discusses several relevant examples which further illustrate the power and the limitations of our criterion.

Keywords:  Teichm├╝ller geodesic flow, moduli space of abelian differentials, Kontsevich--Zorich cocycle, non-uniform hyperbolicity.
Mathematics Subject Classification:  Primary: 30F30, 32G15, 32G20, 57M50; Secondary: 14D07, 37D25.

Received: October 2010;      Revised: April 2011;      Available Online: July 2011.