Mathematics Department Colloquium
Spring 2005

Date: Thursday, February 24
Time: 4:00 p.m.
Location: 215 Thomas Building
Name: Giovanni Forni
Affiliation: Northwestern University
Title: Renormalization and quantitative equidistribution for parabolic flows

Abstract: A flow is called parabolic if nearby orbits diverge with at most polynomial speed with time. Examples of such flows include billiards in polygons, conservative flows with saddle singularities on surfaces (related to interval exchange transformations), horocycle flows and nilflows. For the typical parabolic flow all trajectories tend to equidistribution and for applications, for instance to number theory, it is important to know the equidistribution speed (for smooth functions). In this talk we will describe an approach to this questions based on the introduction of an appropriate renormalization dynamics and on the study of the cohomological equation and of invariant distributions of the flow. The renormalization dynamics is hyperbolic and can be studied with tools of hyperbolic theory such as Lyapunov exponents. For instance, in the case of conservative flows on surfaces the renormalization is given by the Teichmueller flow on the moduli space of holomorphic differentials and for horocycle flows by the corresponding geodesic flow. The cohomological equation can be studied by tools of Fourier analysis/representation theory, although in some cases a dynamical approach is also possible. Interesting applications to number theory come from the study of nilflows.