Existence and Nonexistence of Invariant Curves of Coin Billiards

The candidate solution accurately captures the paper’s three main thrusts: (i) KAM existence in two perturbative regimes (near-circular tables and small height), aligning with Theorems 2 and 3, (ii) nonexistence of invariant essential curves near the boundary for noncircular tables at large height, including the explicit ℓ0 = -3/min ρ'' threshold (Theorem 9), and (iii) rigidity that a full foliation by essential invariant curves forces the table to be a disc (Theorem C). It uses standard symplectic/normal-form/KAM arguments consistent with the paper’s approach, though with some notational and minor quantitative deviations (e.g., the disc formula and O(ε) vs O(ε^2) sizes in one normal form). Overall, the paper’s statements and proofs are sound and the model’s outline is substantively correct though not identical in presentation or normalization choices (see formulation of the coin map and near-boundary expansion, the KAM regimes, and the large-height threshold in the paper: , , , , , ).