On infinitely many foliations by caustics in strictly convex open 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 an open billiard whose boundary is a strictly convex embedded (non-closed) curve γ. We prove that there exists a domain adjacent to γ from the convex side and a C∞-smooth foliation of its union with γ whose leaves are γ and non-closed caustics . This generalizes a previous result by R.Melrose on existence of a germ of foliation as above. We show that there exist a continuum of above foliations by caustics whose germs at each point in γ are pairwise different. We prove a more general version of this statement for γ being an (immersed) arc. It also applies to a billiard bounded by a closed strictly convex curve γ 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∞-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.

Mots clés: billiards; caustics; integrability

Domaines: Systèmes dynamiques [math.DS]

Alexey Glutsyuk. On infinitely many foliations by caustics in strictly convex open billiards. Ergodic Theory and Dynamical Systems, In press, ⟨10.1017/etds.2023.42⟩. ⟨hal-03438666v3⟩