Convergence of rare event point processes to the Poisson process for planar billiards

We show that for planar dispersing billiards the distribution of return times is, in the limit, Poisson for metric balls almost everywhere w.r.t. the SRB (Sinai–Ruelle–Bowen) measure. Since the Poincaré return map is piecewise smooth but becomes singular at the boundaries of the partition elements, recent results on the limiting distribution of return times cannot be applied, as they require the maps to have bounded second derivatives everywhere. We first prove the Poisson limiting distribution assuming exponentially decaying correlations. For the case where the correlations decay polynomially, we induce on a subset on which the induced map has exponentially decaying correlations. We then prove a general theorem according to which the limiting return times statistics of the original map and the induced map are the same.