C0-GAP BETWEEN ENTROPY-ZERO HAMILTONIANS AND AUTONOMOUS DIFFEOMORPHISMS OF SURFACES

The paper proves that for any surface Σ there exists an entropy-zero Hamiltonian g that is not a C0-limit of autonomous diffeomorphisms (Theorem 1), by constructing g as a composition ϕ ∘ ψ supported in an embedded annulus, proving integrability via an invariant lamination, and deriving a contradiction for any autonomous h that is C0-close using a fixed-point index argument on a carefully chosen disk inside the annulus (Lemma 4 and the subsequent index argument) . The candidate solution mirrors this construction and the index-based obstruction, differing mainly in technical presentation: it frames the contradiction directly with the degree of x ↦ φ(x) − x on ∂D (rather than the paper’s use of h2 and a radius r0.2), and it sketches the invariant first integral in terms of preserving a lamination. Integrability implying zero entropy is consistent with the paper’s discussion referencing Katok . Minor gaps in the candidate (e.g., the commuting comment and the precise construction of the first integral) are easily repairable and do not change the substance. Overall, both are correct and use substantially the same embedded-annulus + index-obstruction strategy.