On the behavior of trajectories of scattering billiards on the flat torus

Properties of trajectories of scattering billiards on the flat two-dimensional torus are considered. A Riemann surface is associated with such a billiard in a natural way, while the billiard flow lifts in a natural way to the manifold of linear elements of this Riemann surface. The lift of the billiard flow is a multivalued flow. The trajectories of the billiard flow are lifted from the Riemann surface to the Lobachevsky plane, and properties of the “exponential billiard mapping” are studied. On the absolute circle of the Lobachevsky plane there arise Cantor sets – “reachable sets” of the billiard flow. The metric characteristics – the Hausdorff dimension and the Pontryagin–Schnirelmann orders – of these Cantor sets are invariants of the original billiard system. The topological entropy of the billiard flow on the Riemann surface is estimated in terms of these invariants.