Free Path of Billiards with Flat Points

In this paper we study a special family of Lorentz gas with infinite horizon. The periodic scatterers have C3 smooth boundary with positive curvature except on finitely many flat points. In addition there exists a trajectory with infinite free path and tangentially touching the scatterers only at some flat points. The singularity set of the system is analyzed in detail. And we prove that the free path is piecewise Hölder continuous with uniform Hölder constant. In addition these systems are shown to be non-uniformly hyperbolic; local stable and unstable manifolds exist on a set of full Lebesgue measure; and the stable and unstable holonomy maps are absolutely continuous.