Caustics for inner and outer billiards

With a plane closed convex curve,T, we associate two area preserving twist maps: the (classical) inner billiard inT and the outer billiard in the exterior ofT. The invariant circles of these twist maps correspond to certain plane curves: the inner and the outer caustics ofT. We investigate how the shape ofT determines the possible location of caustics, establish the existence of open regions which are free of caustics, and estimate fro below the size of these regions in terms of the geometry ofT.Partially supported by NSF.Partially supported by NSF Grant DMS 9017995.