Differentiability of invariant circles for strongly integrable convex billiards (Q372577)

The author takes any convex billiard table in the plane, with \(C^2\) boundary. It's phase space consists of the unit vectors directed inside the billiard table from points of its boundary. This phase space is transformed to itself by allowing a billiard ball to travel in the given direction and begin bouncing off in a new direction. A billiard table is integrable if a full measure subset of the phase space is foliated by invariant closed curves. If the limiting leaves of this foliation are either closed curves or discrete points, then the author proves that the set of phase points with irrational slope is foliated into invariant \(C^1\) circles. He also proves that if the set of phase points with rational slope is either foliated in invariant circles or contains two invariant circles which are both \(C^1\) but for finitely many points.