Deviation of ergodic averages for area-preserving flows on surfaces of higher genus

TLDR: In this paper, a substantial part of a conjecture of Kontsevich and Zorich on the Lyapunov exponents of the Teichmuller geodesic flow on the deviation of ergodic averages for generic conservative flows on higher genus surfaces was proved.

Abstract: We prove a substantial part of a conjecture of Kontsevich and Zorich on the Lyapunov exponents of the Teichmuller geodesic flow on the deviation of ergodic averages for generic conservative flows on higher genus surfaces. The result on the Teichmuller flow is formulated in terms of a (symplectic) cocycle on the real cohomology bundle over the moduli space of holomorphic differentials introduced by Kontsevich and Zorich. We prove that such a cocycle is non-uniformly hyperbolic, that is, all of its Lyapunov exponents are different from zero. In particular, the number of strictly positive exponents is equal to the genus of the surface. From this theorem we derive that ergodic integrals of smooth functions for generic area-preserving flows on higher genus surfaces grow with time according to a power-law asymptotics with a number of terms equal to the genus of the surface and stricltly positive exponents equal to the non-negative Lyapunov exponents of the Kontsevich-Zorich cocycle. In particular, for conservative flows on surfaces of higher genus, the deviation of ergodic averages for a generic smooth function obeys a power law with a strictly positive exponent and, consequently, the Denjoy-Koksma inequality does not hold. The derivation of the deviation theorem relies in a fundamental way on the notion of invariant distribution for flows on surfaces and the related notion of basic current for the orbit foliation.