Asymptotic behaviour of ergodic integrals of `renormalizable' parabolic flows

Abstract

Ten years ago A. Zorich discovered, by computer experiments on interval exchange transformations, some striking new power laws for the ergodic integrals of generic non-exact Hamiltonian flows on higher genus surfaces. In Zorich's later work and in a joint paper authored by M. Kontsevich, Zorich and Kontsevich were able to explain conjecturally most of Zorich's discoveries by relating them to the ergodic theory of Teichm\"uller flows on moduli spaces of Abelian differentials. In this article, we outline a generalization of the Kontsevich-Zorich framework to a class of `renormalizable' flows on `pseudo-homogeneous' spaces. We construct for such flows a `renormalization dynamics' on an appropriate `moduli space', which generalizes the Teichm\"uller flow. If a flow is renormalizable and the space of smooth functions is `stable', in the sense that the Lie derivative operator on smooth functions has closed range, the behaviour of ergodic integrals can be analyzed, at least in principle, in terms of an Oseledec's decomposition for a `renormalization cocycle' over the bundle of `basic currents' for the orbit foliation of the flow. This approach was suggested by the author's proof of the Kontsevich-Zorich conjectures and it has since been applied, in collaboration with L. Flaminio, to prove that the Zorich phenomenon generalizes to several classical examples of volume preserving, uniquely ergodic, parabolic flows such as horocycle flows and nilpotent flows on homogeneous 3-manifolds.