On infinitely many foliations by caustics in strictly convex open billiards

Abstract

Reflection in strictly convex bounded planar billiard acts on the space of oriented lines and preserves a standard area form. A caustic is a curve $C$ whose tangent lines are reflected by the billiard to lines tangent to $C$. The famous Birkhoff Conjecture states that the only strictly convex billiards with a foliation by closed caustics near the boundary are ellipses. By Lazutkin's theorem, there always exists a Cantor family of closed caustics approaching the boundary. In the present paper we deal with an open billiard, whose boundary is a strictly convex embedded (non-closed) curve $\gamma$. We prove that there exists a domain $U$ adjacent to $\gamma$ from the convex side and a $C^\infty$-smooth foliation of $U\cup\gamma$ whose leaves are $\gamma$ and (non-closed) caustics of the billiard. This generalizes a previous result by R.Melrose, which yields existence of a germ of foliation as above at a boundary point. We show that there exists a continuum of above foliations by caustics whose germs at each point in $\gamma$ are pairwise different. We prove a more general version of this statement in the cases, when $\gamma$ is just an arc, and also when both $\gamma$ and the caustics are immersed curves. It also applies to a billiard bounded by a closed strictly convex curve $\gamma$ and yields infinitely many "immersed" foliations by immersed caustics. For the proof of the above results, we state and prove their analogue for a special class of area-preserving maps generalizing billiard reflections: the so-called $C^{\infty}$-lifted strongly billiard-like maps. We also prove a series of results on conjugacy of billiard maps near the boundary for open curves of the above type.