NON-DENSITY RESULTS IN HIGH DIMENSIONAL STABLE HAMILTONIAN TOPOLOGY

The paper proves (1) non-density of stable hypersurfaces in dim W ≥ 8 (Theorem 1) and (2) in every regular stable homotopy class on a (2n+1)-manifold with n≥2 there is a SHS that cannot be C2-perturbed to a non-degenerate one (Theorem 2). The core ingredients are: a non-R-covered, volume-preserving Anosov flow whose associated 2-form admits no stabilizer and remains non-stabilizable under C1-small perturbations (Lemma 15) ; a local symplectic embedding and insertion of a 3D NHIM with Anosov dynamics into any hypersurface in dim ≥8 (Propositions 18, 21, 22) ; persistence of NHIMs (Theorem 9) ensuring the obstruction survives C3-approximation, and explaining why C2 is insufficient (Remark 24) ; and, for (2), coupling functions (Lemma 7), a degeneracy criterion (Lemma 28), and a local model that robustly forces a periodic orbit on a regular level set (Proposition 29) . The candidate solution closely mirrors these steps, including the Anosov kernel, NHIM insertion, the C3-threshold reasoning, and the invariant tori with rotation-number continuity. Minor issues: a slight misnumbering of statements and an imprecise phrasing of “coupling functions” (the paper uses functions fi derived from dλ and powers of ω), but these do not affect the logic. Overall, both are correct and follow substantially the same proof.