Billiard tables with rotational symmetry

The paper states and proves four main theorems: Birkhoff (circle), Outer (ellipse), Symplectic (ellipse), and a Minkowski criterion yielding extra ak-fold symmetry with necessity and sufficiency, all under a k-fold linear symmetry and the existence of a rotational invariant curve of k-periodic points. These are clearly formulated as Theorems 1–4, and proved via a careful parametrization, the commutation T∘A = A∘T along the invariant circle, and Birkhoff/Aubry–Mather tools; see the theorem statements and the section headers for Birkhoff, Outer, Symplectic and Minkowski billiards respectively . For Birkhoff, the authors deduce a constant bounce angle and then invoke Cyr’s number-theoretic lemma to force a circle . For Outer and Symplectic, they reduce to trigonometric/Fourier identities showing only the n=±1 modes survive, hence an ellipse . For Minkowski, they prove that the bounce points lie on the A-orbit (a power B=A^m) and that the action on that invariant curve is constant, implying that g_K(Bγ(t)−γ(t)) is constant; from there they establish the ak-symmetry and its sufficiency by a concrete factorization B−I = 2 sin(π m/k) R_{π/2+π m/k} and a commutation argument . In contrast, the candidate solution contains substantial gaps in the Minkowski part. It asserts that g_K(A^r x − x) ≡ c on ∂K implies g_K(A^r x − x) = c g_K(x) for all x (this scaling extension is fine), defines B = (A^r − I)/c as a g_K-isometry, and then claims—by comparing with the Euclidean identity A^r − I = 2 sin(π r/k) R_{π/2+π r/k}—that necessarily c = 2 sin(π r/k) and B = R_{π/2+π r/k}. This “comparison” is unjustified: the Euclidean factor 2 sin(π r/k) is norm-specific and does not follow from the Minkowski scaling relation; one cannot conclude that B equals that Euclidean rotation, nor that B has the stated finite order without additional argument. The paper circumvents this by proving invariance under an explicit finite-order linear map via a separate lemma, not by identifying B with a Euclidean rotation . The sufficiency direction in the model also presumes the cyclic isometry group contains D_r = R_{π r/k/} J and that A^r − I = 2 sin(π r/k) D_r, which again mixes Euclidean rotation identities with general Minkowski isometries without proof. Additionally, for the Outer case the model relies on an unstated “outer–symplectic duality” (not used by the paper), citing Dual billiards, whereas symplectic billiards were introduced later; the paper provides a direct Outer proof instead . The Birkhoff and Symplectic parts of the model are essentially correct (different proofs) and align with the paper’s conclusions, but the Minkowski necessity/sufficiency steps are flawed.