Differentiability of invariant circles for strongly integrable convex billiards

Abstract

Let $C$ be a closed convex curve of class $C^2$ in the plane. We consider the domain bounded by $C$ a billiard table. Assume that the convex billiard of $C$ is integrable and satisfies a certain property. The property is that the limiting leaves are either closed curves or discrete points in the phase space. Then the set of points with irrational slopes make invariant circles of class $C^1$. If the sets of points with rational slopes do not make invariant circles, then they contains two invariant circles such that they are of class $C^1$ except at finitely many points in $C$.