On infinitely many foliations by caustics in strictly convex open billiards

TL;DR

This paper demonstrates the existence of infinitely many smooth foliations by caustics in open strictly convex billiards, extending previous results and introducing new classes of such foliations near the boundary.

Contribution

It proves the existence of a continuum of distinct caustic foliations near the boundary of open convex billiards, generalizing earlier germ-level results to entire neighborhoods and immersed curves.

Findings

Existence of smooth caustic foliations near boundary curves.

Infinitely many pairwise different foliations for open convex billiards.

Generalization to immersed curves and closed convex boundaries.

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.