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

The paper proves the existence of a C∞ foliation of a collar by billiard caustics for any strictly convex C∞ arc γ, and moreover, a continuum of such foliations with pairwise distinct germs along γ (Theorem 1.2 and the addendum to Theorem 1.4). The proof proceeds via the class of C∞-lifted strongly billiard-like maps, establishing a normal form with flat remainders F̃(τ,φ) = (τ + φ + flat(φ), φ + flat(φ)) (equation (1.11)) and constructing an invariant function φ̃ = φ + flat(φ), whose level sets give the invariant foliation (Theorems 1.15 and 1.18, and §2.7 for the billiard reduction) . By contrast, the candidate solution assumes a stronger, unproved C∞ symplectic conjugacy to the exact model F0(t,z) = (t + √z, z) and then pulls back the horizontal foliation {z = const}. The paper does not claim such an exact conjugacy; it works with the flat-error normal form (1.11) and an invariant first integral to produce the foliation. Thus, while the candidate’s final conclusions match the paper’s statements, the key step (global conjugacy to F0 up to the boundary) is not established in the paper and is not obviously valid in general.