SYMPLECTIC BILLIARDS FOR PAIRS OF POLYGONS

The paper’s Theorem 3.5 and its proof are sound: periodicity on P\C#, almost-everywhere periodicity on Pmax when C is closed, and a sharp global period bound when C is finite, all follow from the tile/phase-rectangle framework and a return map of order at most 4 (via Lemma 2.21) together with a counting argument for arc-wise components of (P−×P+)⊔(P+×P−)\C# . The candidate solution correctly recovers parts (II)–(III) by a piecewise-affine “rectangle permutation” argument compatible with Lemma 2.19 (ϕ(x,y)=(y,ax+b) on each fixed triple of edges) and achieves the same period bounds . However, its proof of (I) relies on an unjustified finiteness claim: it asserts that a neighborhood U of an arbitrary orbit meets only finitely many C-grid lines if C accumulates only at vertices, hence the orbit lies in a finite cycle of rectangles. This is not valid in general because an orbit may approach a vertex arbitrarily closely, and the C-grid can have infinitely many lines accumulating at that vertex while still avoiding it; thus U can intersect infinitely many grid lines. The paper avoids this pitfall by constructing phase rectangles via “closest point to the left” and invoking the tile return map of bounded order, which remains valid under the (I) hypothesis that all limit points of C lie at vertices .