Local Rigidity for Hyperbolic Toral Automorphisms

The paper’s Theorem 3.2 states precisely the claim under review: for a weakly irreducible hyperbolic toral automorphism L there exists r(L) so that any C∞ f sufficiently Cr(L)-close to L has the property that if some conjugacy H (or H−1) lies in W^{1,q}(Td) with q>d, then any conjugacy is C∞ . The paper first upgrades Sobolev regularity to C1+β via Theorem 2.10 by differentiating the conjugacy equation and applying their measurable-to-Hölder cocycle rigidity (Theorem 2.9) , and then achieves C∞ using a new KAM-like iterative scheme solving a twisted linearized equation with Fourier methods and a Lyapunov splitting reduction . The candidate solution mirrors the same program up to C1+ (Sobolev embedding, measurable cocycle cohomology, Journé’s lemma), then diverges by invoking non-stationary normal forms and canonical perturbation theory to bootstrap to C∞. While the proof techniques differ from the paper’s scheme, the model’s outline aligns with established rigidity tools and reaches the same result; the paper additionally notes Walters’ centralizer fact that any two conjugacies differ by an affine map commuting with L, hence have the same regularity . The main caveat is that the model assumes fiber-bunching and finite-order normal-form ‘narrow band’ inequalities without fully quantifying them, whereas the paper secures the needed regularization via its Theorem 2.9 and an explicit iterative argument. Overall, the paper’s argument is complete and correct, and the model’s solution is a plausible alternative route with standard ingredients but fewer details on quantitative hypotheses.