Invariant distributions in parabolic dynamics

In the past two decades L. Schwartz theory of distributions has become an essential tool in the study of smooth dynamical systems. In this talk we have focused on two kinds of applications to parabolic systems: smooth rigidity and effective equidistribution. Parabolic systems are zero entropy systems characterized by polynomial divergence of nearby orbits with time. Examples include unipotent flows on quotient of semisimple groups, in particular horocycle flows for hyperbolic surfaces, nilflows and translation flows on surfaces. The rigidity problem asks whether systems which are topologically conjugated are necessarily smoothly conjugated. This problem is motivated by the results of M. Herman and J.-C. Yoccoz on circle diffeomorphisms, but is wide open in higher dimensions. We review results and related conjectures on cohomological equations for flows and propose a general conjecture that states that the structure of the space of its invariant distributions determines whether a smoothly stable system is rigid. Effective equidistribution consists in bounds on the speed of convergence of ergodic averages (of smooth functions). We present two results of effective equidistribution for homogenous flows: the first on twisted horocycle flows, the second on a class of nilflows, with applications to questions in analytic number theory. Our method is based on the analysis of the scaling of Sobolev norms of invariant distributions under scaling of the homogeneous structure.