Title: New analytical tools for dynamics with singularities, including
Sinai billiards


Abstract: In the past 15 years, tools from analysis, in particular new
Banach spaces of anisotropic  distributions on manifolds, have allowed substantial progress in  dynamical systems.

After briefly explaining how a spectral gap
for a transfer operator furnishes ergodic information, we shall focus
on Sinai billiards maps and flows. These  natural but technically challenging systems are uniformly hyperbolic and volume preserving - however grazing orbits give  rise to singularities.
New analytic tools recently allowed us to obtain exponential mixing
for finite horizon Sinai billiard flows (with M. Demers and
C. Liverani), and the natural volume. We shall
finish by discussing ongoing work (with M. Demers)
on other Gibbs states (including the measure of maximal
entropy).