Regularity of local manifolds in dispersing billiards

This work is devoted to 2D dispersing billiards with smooth bound- ary, i.e. periodic Lorentz gases (with and without horizon). We revisit several fundamental properties of these systems and make a number of improvements. The necessity of such improvements became obvious during our recent studies of gases of several particles [CD]. We prove here that local (stable and unstable) manifolds, as well as singularity curves, have uniformly bounded derivatives of all orders. We establish sharp estimates on the size of local manifolds, on distortion bounds, and on the Jacobian of the holonomy map.