TANGENT SPACE OF THE STABLE AND UNSTABLE MANIFOLD OF ANOSOV DIFFEOMORPHISM ON 2-TORUS

The paper’s Theorem 2 states that for the Anosov perturbation S_ε on T^2 and the structural conjugacy H_ε, the limit v_ε(ψ) = lim_{t→0} V_+(ψ,t)/V_-(ψ,t) exists (with V defined from H_ε(ψ+tv_−)−H_ε(ψ)) and then develops a formal-series/derivative-tree machinery to justify and compute it; see the statement and proof roadmap around Theorem 2 and Remark 1 . The candidate solution instead gives a short geometric proof: by structural stability H_ε maps the straight stable line to the C^r (indeed analytic) stable manifold W^s_ε and the ratio V_+/V_- is the slope of secants of a C^1 graph, hence converges to the tangent slope. This relies on standard Anosov stable manifold theory (the leaves are as smooth as the diffeomorphism) and persistence/continuity of the stable bundle under small C^1 perturbations, which the paper itself recalls in Theorem 1 and Proposition 1 (iii) . The paper’s proof is more elaborate (to extract an explicit expansion); there is a minor expositional gap in the final convergence step where the authors say it “could be easily done” (following Proposition 4 and Proposition 5) without fully writing it out , but the intended estimates are standard and the overall argument is sound. Hence both are correct; the model offers a simpler, different proof.