ON THE DYNAMICS OF ROTATING RANK-ONE STRANGE ATTRACTORS FAMILIES

The paper’s Theorem D asserts that for (α2, δ2) in the interior of an Arnold tongue Up/q and for a positive-measure set of ε1 with ε2 small, the three-dimensional family F(ε1, ε2) exhibits an irreducible strange attractor contained in the unstable manifold of a q-periodic saddle p⋆a, and that this attractor shadows S1 × {1} × OrbF3(ts1). The paper proves this by reducing to two-dimensional maps on the attracting t-sheets Ωj (t = tsj), composing Gq ∘ ··· ∘ G1 to obtain a C3 ε2-perturbation of a planar rank-one map T(ε1,0), and then transferring the rank-one attractor back to F; see the statement of Theorem D and its proof via (9.1)–(9.2) and Lemma 9.1 . The assumptions (H1)–(H6) and the Arnold tongue description for F3 underpin the construction . The planar rank-one theory is invoked to obtain an SRB measure and shadowing by y = 1 for the 2D subsystem, and then a dense orbit with a positive exponent in W u(p⋆) for F, concluding Theorem D . The candidate solution reaches the same main conclusion but via a different (and plausible) mechanism: it constructs a forward-invariant trapping region over the attracting t–cycle, invokes a strong-stable foliation in the (y, t) directions for P = F q, and reduces to a 1D quotient in x fitting the Wang–Young rank-one scheme. This differs from the paper’s restriction-to-Ωj and composition approach. The core steps—existence of a stable q–cycle for F3 in int(Up/q), strong contraction in (y, t), a rank-one admissible quotient with nondegenerate critical set, and positive-measure parameter selection—are consistent with the paper’s hypotheses and logic. Two overstatements appear in the candidate solution: (i) it claims uniform expansion along the unstable direction on the attractor, whereas the paper’s rank-one attractors are nonuniformly hyperbolic (with criticalities) and only guarantee a positive Lyapunov exponent, not uniform expansion ; (ii) it posits a quantitative O(ε1^θ) + O(ε2) Hausdorff shadowing rate that the paper does not establish, though the qualitative shadowing statement is part of Theorem D’s description . These points do not affect the main theorem’s validity but go beyond what is proved in the paper.