A Franks' lemma for convex planar billiards

Let. be an orbit of the billiard flow on a convex planar billiard table; then the perpendicular part of the derivative of the billiard flow along. is a symplectic linear map DP. This paper contains a proof of the following Franks' lemma for a residual set of convex planar billiard tables: for any closed orbit, the map DP can be perturbed freely within a neighbourhood in Sp(1) by a C-2-small perturbation in the space of convex planar billiard tables.