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

Giovanni Forni

doi:10.3934/jmd.2011.5.355

Article Contents

2011, Volume 5, Issue 2: 355-395. Doi: 10.3934/jmd.2011.5.355

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

Giovanni Forni^1,

1.
Department of Mathematics, University of Maryland, College Park, MD 20742-4015

Received: September 30, 2010

Revised: March 31, 2011

Published: June 2011

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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.

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Mathematics Subject Classification: Primary: 30F30, 32G15, 32G20, 57M50; Secondary: 14D07, 37D25.

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