ON THE PERSISTENCE OF PERIODIC TORI FOR SYMPLECTIC TWIST MAPS AND THE RIGIDITY OF INTEGRABLE TWIST MAPS

The paper’s Theorem 1 shows that, under strong positivity and analyticity, the parameter set of ε producing a Lipschitz Lagrangian (m,n)-periodic graph is either all of R or consists of isolated points, and—adding bounded rate with non-constant G—is finite. The proof proceeds by complexifying, analyzing the complex submanifold of radially transformed points R* and deploying two holomorphic defect maps Δ and χ to extend periodic/Lagrangian properties by analyticity; uniqueness and invariance of periodic Lagrangian graphs are also invoked. All these ingredients appear explicitly in the text (definitions of R, R*, Δ, χ, Lemma 2.7, Lemma 2.8, Appendix A and C) and match the candidate’s approach, which uses the same scheme (complexification; holomorphic identity; uniqueness/invariance; bounded-rate compactness/no-graph at large |ε|). Minor issues in the candidate solution are only bibliographic (mis-numbered lemmas) and a small terminology slip distinguishing R vs P, but the logical content and conclusions agree with the paper’s argument. Hence both are correct and substantially the same proof. See Theorem 1 and its proof sketch (complexification via R*, Δ, χ) and the bounded-rate finiteness argument (Proposition C.2) in the paper .