Dynamics of the semigroup of contractive automorphisms of Banach spaces

The paper proves Theorem 4.2 by a clean curvature-comparison argument using the tailored linear maps Lab_ε, a quantitative curvature amplification lemma (Lemma 4.5), and a uniqueness-of-intersection lemma for C2 plane curves (Lemma 4.3). This yields a rigorous local-to-global barrier ensuring L[S] lies inside the unit ball and meets S only at the target point, hence a contractive automorphism close to the identity exists. The candidate solution reproduces the local second-order analysis (via the Hessian of the norm) and constructs near-identity maps that align x, y and tangents, but its key ‘local-to-global’ step is not justified: the argument that a strict local maximum at x forces the global maximum on S, solely from L→I, is unproven and in general false without an additional curvature-based barrier. Thus the paper’s proof is correct and complete, while the model’s solution is incomplete.