Department Colloquium
Department of Mathematics
Princeton University
February 21st, 2001
On the smooth ergodic theory of some examples
of parabolic flows
We will present recent results on the behaviour of ergodic averages of
smooth functions for two examples of `parabolic' conservative flows:
generic area-preserving flows (with saddle-like singularities) on
higher genus surfaces and horocycle flows on (compact) surfaces of
constant negative curvature. We prove that the deviation of ergodic
averages from the leading behaviour determined by the ergodic theorem
exhibits a power-law decay controlled by invariant distributions. In
the case of flows on higher genus surfaces this result was part of a
series of conjectures by M.Kontsevich and A.Zorich. The proofs are
based on the analysis of the hyperbolicity properties of the appropriate
`renormalization' dynamics, related to the Teichmuller flow on the
moduli space in the case of flows on higher genus surfaces and to
the geodesic flow in the case of horocycle flows.