CANTOR SET STRUCTURE OF THE WEAK STABILITY BOUNDARY FOR INFINITELY MANY CYCLES IN THE RESTRICTED THREE-BODY PROBLEM

Under Assumptions 3.10/3.11 (transverse invariant-manifold intersections; Hypothesis A) the paper correctly identifies a Smale–Birkhoff horseshoe CC on each section Sθ and states W1C(θ)=CC∩Λ∩Sθ (Cor. 3.13) . It then claims Lemma 3.14 (“W1C(θ) is a Cantor set”) from “W1C(θ) is closed” (Property 3.9, built on Assumption 3.8) and “W1C(θ)⊂CC” , but “closed subset of a Cantor set” need not be Cantor without establishing perfectness/transversal slicing. Thus the paper’s proof of the Cantor property is logically incomplete, even though the overall result is very plausible. The model supplies the missing hyperbolic-sets argument: via local product structure of CC and transversality of Λ∩Sθ to the stable foliation, the slice CC∩Λ∩Sθ is Cantor; it also re-derives Wn(θ)=WA n(θ) in line with the paper’s Lemma 3.12 (based on [7]) . In short, the paper’s main identification and use of a horseshoe are sound under its assumptions, but the final step to “Cantor” is not justified as written; the model’s solution fixes this gap.