A FIXED POINT CURVE THEOREM FOR FINITE ORBITS LOCAL DIFFEOMORPHISMS

The paper proves the Fixed Point Curve Theorem in C^2: for any local biholomorphism with finite orbits, some iterate has a germ of analytic curve contained in its fixed set; this is stated in the introduction and established via a semianalytic analysis of the periodic set and Theorem 4 on connected components of Per(F) (which become components of Per_m(F) and are complex-analytic of positive dimension in open balls) , with a key stabilization step Lemma 4.6 . The paper also records the standard spectral restriction |λ|=1 from a holomorphic stable manifold argument (cf. their Corollary 5, referenced in the discussion around Theorem 3) , and uses the fixed-curve result to deduce that at least one eigenvalue is a root of unity (proof of Theorem 3) . By contrast, the model’s “solution” merely cites the very Fixed Point Curve Theorem it is supposed to establish, rendering it circular; its tangent-to-identity consistency check aligns with the paper’s discussion of Abate’s parabolic-curve results but does not supply a proof of the general case . Hence: paper correct; model not a valid proof.