ESSENTIAL DYNAMICS IN CHAOTIC ATTRACTORS

The paper’s Theorem 1.1 assumes only that on a cross-section S1 the first-return map g|_I is topologically conjugate to a Smale or Fake horseshoe and extends continuously to S1, then proves F has infinitely many periodic orbits, with generic persistence and knot-type invariance (via a relative isotopy from G to F, a Bestvina–Handel/Pseudo-Anosov embedding, and Orbit Index arguments) . The candidate solution incorrectly upgrades these purely topological hypotheses to uniform hyperbolicity of a suspended set Λ_G and then applies structural stability to continue Λ_G along the homotopy to F. That step is unjustified here: g need not be a diffeomorphism on I, and the presence of “Fake horseshoe” dynamics and of Möbius index-0 periodic orbits in smooth suspensions of the Smale horseshoe (Appendix B) shows the suspended set need not be a uniformly hyperbolic basic set for the flow . Consequently, the model’s blanket claim that all these orbits persist for every small C^3 perturbation and preserve knot type (without genericity) overreaches the paper’s carefully qualified generic statement.