Finite Periodic Data Rigidity For Two-Dimensional Area-Preserving Anosov Diffeomorphisms

The paper proves a uniform, quantitative finite–periodic-data rigidity theorem on T^2 via a concrete “weighted holonomy” construction of h_N, leafwise C^{1+α} regularity plus Journé, and an effective equidistribution theorem for periodic points to the SRB measure to get d_{C^0}(h,h_N) ≤ C_0 λ_0^N and d_{C^1}(f,f_N) ≤ C_1 λ_1^N, with constants uniform over a C^2-bounded family U. These ingredients appear explicitly in Theorem 1.1 and the Section 3–6 roadmap (weighted holonomy to ensure continuity across stable leaves, and effective equidistribution for Lipschitz observables) . By contrast, the candidate solution relies crucially on an unproven “finite quantitative Livšic” statement for diffeomorphisms—producing φ = u_N∘f − u_N + r_N with ||r_N|| ≤ Cρ^N from only S_N φ(p)=0 on Fix(f^N)—and then attempts a straightforward holonomy projection without the weighted averaging that the paper explicitly introduces to fix transverse discontinuity issues. The finite Livšic step is nonstandard here and not justified by the cited flow result; and omitting the weighted holonomy leaves a genuine gap the paper itself highlights (the naive leafwise definition need not extend continuously to T^2) .