ON THE ACCUMULATION OF SEPARATRICES BY INVARIANT CIRCLES

The paper proves, under (H1)–(H4), that for sufficiently small |ε| there is a positive-measure set of invariant circles accumulating the separatrix, and with higher regularity many are KAM circles (Theorem 2.1) . Crucially, the authors state that these circles cannot be obtained directly via a classical KAM approach; instead they construct a first-return map, normalize to a standard annulus, and then rescale to a small Cr perturbation of an integrable twist map before applying Moser–Rüssmann (Sections 4–6) . The candidate solution, by contrast, asserts uniform smallness in action–angle coordinates on arbitrarily thin annuli and applies KAM directly, bypassing the renormalization and rescaling steps. This contradicts the paper’s key technical insight and overlooks the unbounded logarithmic shear present in the first-return normal form lε(v)=σ0,N(v)+n̂ε(u,v)q′ε(v)−ln v (which drives the need to renormalize) . The paper’s argument is careful and complete (normal forms: Proposition 2.1; normalization: Lemma 5.3; renormalization f̄ε with l(y)=σ(y)−ln y; rescaling to f̊ε,n; application of Moser–Rüssmann and Herman–Yoccoz) . The model’s proof omits the essential renormalization and provides no rigorous control showing that the direct KAM smallness thresholds are met uniformly near the separatrix.