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, I 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 allowed us (with M. Demers and
C. Liverani) to obtain exponential mixing
for the natural volume and finite horizon Sinai billiard flows.
With M. Demers, we are now constructing the measure of
maximal entropy for the billiard map. I will try to
compare the two results (without entering in too much
technical details!).