A GEOMETRIC CRITERION FOR THE NONUNIFORM HYPERBOLICITY OF THE KONTSEVICH-ZORICH COCYCLE

作者：Forni Giovanni^*

来源：Journal of Modern Dynamics, 2011, 5(2): 355-395.

DOI：10.3934/jmd.2011.5.355

• Teichmuller geodesic flow
• moduli space of abelian differentials
• Kontsevich-Zorich cocycle
• non-uniform hyperbolicity

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.

• 出版日期2011-4