Pär Kurlberg: A Poincaré section for horocycle flows: escape of mass

Abstract: Motivated by a hyperbolic analog of the Lester-Wigman "vanishing area correlations"-conjecture for euclidean lattice points we investigate the dynamical properties of a natural choice of a Poincare section, associated with H/SL(2,Z), and the horocycle flow on the upper half plane H. Since the horocycle *flow* is mixing, one might hope for an easy proof of vanishing area correlations by showing that the Poincaré map is mixing. However, not only is the Poincare map non-mixing; even equidistribution/ergodicity breaks down badly due to escape of mass. Amusingly, we can still show vanishing of area correlations (but "for the wrong reason".)