On Length Spectrum Rigidity of Dispersing Billiard Systems

The paper proves two main results: (A) equality of the cyclicity‑2 marked length data ℓm,n is equivalent to an analytic conjugacy of the collision maps in a neighborhood of a designated homoclinic orbit, and (B) if in addition D1 ∪ D2 are isometric between the two systems, then the entire billiard tables are isometric. These statements and their proof strategy are clearly stated and executed via a Birkhoff normal form N, a gluing map G, and an explicit asymptotic triangular expansion for ℓm,n with polynomial-in-(m,n) factors multiplying powers of λ2m and λ2n; see Theorem A and Theorem B, and the expansion from Proposition 19 and Corollary 20, including the non-convergence caveat for the substituted series (asymptotic only) . By contrast, the model’s solution asserts a different, overly simplified expansion ℓm,n = (m+n)T* + L∞ + Au(λm) + As(λn) + B(λm, λn) with absolutely convergent power series and without the polynomial m,n prefactors; this contradicts the paper’s triangular structure, its explicit 2m and 2n terms, and the asymptotic (not necessarily convergent) nature of the expansions used in the proof (cf. Proposition 19 and Remark 13) . The model also sketches a generating-function/pole-extraction argument that presumes only simple poles, which is incompatible with the polynomial prefactors (leading to higher-order poles) intrinsic to the paper’s expansion. The model’s step (2) about determining D3 given D1 ∪ D2 and ℓm,n is broadly aligned with the paper’s geometric reconstruction in the proof of Theorem B , but its reliance on the incorrect expansion undermines the claimed justification. Overall, the paper’s results and methods are correct; the model’s proof outline contains nontrivial technical inaccuracies about the expansion and convergence.