On nonconvex caustics of convex billiards

Oliver Knill July 29, 1996 Abstract There are billiard tables of constant width, for which the billiard map has invariant curves in the phase space which belong to continuous but nowhere di erentiable caustics. We apply this to construct ruled surfaces which have a nowhere di erentiable lines of striction. We use it also to get Riemannian metrics on the sphere such that the caustic belonging at least one point on the sphere is nowhere di erentiable. For three dimensional billiards, we nd three dimensional billiard surfaces with nonconvex rough caustics. 1 Convex billiards Describing the long time behavior of a path of a light ray or billiard ball in a convex domain is an interesting mathematical problem. Good starting points in the literature are [21, 31, 6, 33, 27]. In the case, when the domain is a strictly convex planar region, the return map to the boundary T de nes an area preserving map of the annulus A = T [0; 1], where the circle T parametrizes the table with arc length s: the impact point s at the boundary T and the angle of impact de ne the next impact point s1 with angle 1. The map is given by : (s; cos( )) 7! (s1; cos( 1). Noncontractible invariant curves in A can de ne caustics in the interior of the table T . These are curves with the property that a ball, once tangent to stays tangent. The precise relation between the invariant curve and the caustic is discussed in the next section.