Banach spaces and Bernoulli Billiards

It has been a quarter century now that Banach spaces of anisotropic distributions have been introduced to study statistical properties of chaotic dynamical systems via Ruelle transfer operators. This approach gives both new proofs of classical results and new results. The first successes were obtained for smooth hyperbolic dynamics. However, some natural dynamical systems, such as dispersive (Sinai) billiards are not smooth. The singularities cause challenging technical difficulties. We shall survey new results on dispersive (discrete or continuous time) dispersive billiards obtained in the past five years using anistotropic Banach spaces, ending with a very recent construction of the measure of maximal entropy for billiard flows satisfying a condition of sparse recurrence to singularities.

(In this joint work with Carrand and Demers, we obtain Bernoullicity, but no control of the speed of mixing.)