Billiard Tables with Rotational Symmetry

We generalize the following simple geometric fact: the only centrally symmetric convex curve of constant width is a circle. Billiard interpretation of the condition of constant width reads: a planar curve has constant width, if and only if, the Birkhoff billiard map inside the planar curve has a rotational invariant curve of 2-periodic orbits. We generalize this statement to curves that are invariant under a rotation by angle 2π k, for which the billiard map has a rotational invariant curve of k-periodic orbits. Similar result holds true also for outer billiards and symplectic billiards. Finally, we consider Minkowski billiards inside a unit disc of Minkowski (not necessarily symmetric) norm that is invariant under a linear map of order k≥ 3. We find a criterion for the existence of an invariant curve of k-periodic orbits. As an application, we get rigidity results for all those billiards.