# Invariant distributions and scaling in parabolic dynamics

# Invariant distributions and scaling in parabolic dynamics

A smooth dynamical system is often called parabolic if nearby orbits diverge with power-like (polynomial) speed. There is no general theory of parabolic dynamics and a few classes of examples are relatively well-understood: area-preserving flows with saddle singularities on surfaces (or, equivalently, interval exchange transformations) and to a lesser extent 'rational' polygonal billiards; $SL(2,R)$ unipotent subgroups (horocycle flows on surfaces of constant negative curvature) and nilflows. In all the above cases, the typical system is uniquely ergodic, hence ergodic averages of continuous functions converge unformly to the mean. A fundamental question concerns the speed of this convergence for sufficiently smooth functions. In many cases it is possible to prove power-like (polynomial) upper bounds. A unified approach to this problem consists in constructing invariant distributions (in the sense of L. S. Sobolev or L. Schwartz) usually by methods of harmonic analysis and studying how they rescale under an appropriate 'renormalization' scheme. This approach yields quite precise bounds for many of the above examples but often cannot be implemented for lack of an (effective) renormalization. In this talk, after a review of some of the main known results for renormalizable systems, we will present a quantitative equidistribution result for some non-renormalizable nilflows and we will discuss some new ideas we have introduced (in joint work with L. Flaminio) to deal with this problem. Bounds on Weyl sums that can be derived from our results will be discussed.