Billiard tables with rotational symmetry

TL;DR

This paper extends classical geometric and billiard theory results to curves with rotational symmetry, establishing conditions for invariant curves of periodic orbits in various billiard systems and deriving related rigidity results.

Contribution

It generalizes the characterization of constant width curves to those invariant under rotation, linking symmetry to invariant billiard orbits and providing criteria for Minkowski billiards with symmetry.

Findings

Invariant curves exist for k-periodic orbits in symmetric billiards.

Rotational symmetry implies the existence of invariant periodic orbits.

Rigidity results constrain billiard shapes with symmetry and invariance.

Abstract

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 $\frac{2\pi}{k}$, 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 which is invariant under a linear map of order $k\ge 3$. 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.