On infinitely many foliations by caustics in strictly convex non-closed billiards

Alexey Glutsyuk (1)

1 UMPA-ENSL - Unité de Mathématiques Pures et Appliquées

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 billiards whose boundary contains a strictly convex embedded (non-closed) curve $\gamma$. We prove that there exists a domain adjacent to $\gamma$ from the convex side that is $C^{\infty}$-smoothly foliated by non-closed caustics. This generalizes a previous result by R.Melrose, which yields existence of a local foliation by caustics in a neighborhood of 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 case, 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.

Mots clés: billiards; caustics; integrability

Domaines: Systèmes dynamiques [math.DS]

Alexey Glutsyuk. On infinitely many foliations by caustics in strictly convex non-closed billiards. 2021. ⟨hal-03438666v1⟩