Birkhoff attractors for dissipative symplectic billiards

The paper states and sketches a proof of Theorem 3.7—existence, for strong dissipation, of a normally contracted invariant circle that equals the Birkhoff attractor and, for even smaller dissipation, C^{k−1} regularity with C^1 convergence to the zero section—using a cone-field criterion and standard normally hyperbolic theory, referring to earlier work for details. The candidate solution delivers a self-contained graph-transform proof in generating-function coordinates, derives the needed trapping strip and contraction, and invokes normally hyperbolic manifold theory for higher regularity. Apart from minor slips (sign of twist, a minor bound typo, and an implicit lower bound on L11 extended from the zero branch to a small tube), the model’s argument matches the paper’s result and methodology at a comparable level of rigor.