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2204.14135

Topological Shadowing Methods in Arnold Diffusion: Weak Torsion and Multiple Time Scales

Andrew Clarke, Jacques Fejoz, Marcel Guàrdia

correctmedium confidence
Category
Not specified
Journal tier
Specialist/Solid
Processed
Sep 28, 2025, 12:56 AM

Audit review

The paper’s Theorem 2.9 explicitly proves the existence of orbits shadowing a prescribed chain of leaves and establishes the lower bound T ≳ ε^{-(ρ+τ+υ)} for the time to achieve O(1) drift in p, with ρ = max{2σ, 2υ, τ} and the near-integrable error order k ≥ 2(ρ+τ)+1; the same bound is reiterated for the extended system in Theorem 2.11 (equations (11)–(12)) . The proof uses correctly aligned windows, large spectral-gap smoothness of holonomies, the scattering map defined on a homoclinic channel, and a non-uniform twist shearing estimate, matching the model’s method and parameter scalings; see the set-up [A1–A3], the large spectral gap and scattering map construction, and the heuristic/proof outline, as well as the aspect-ratio bookkeeping culminating in Ki = O(ε^{-(ρ+τ)}) and the time lower bound . The model’s solution follows the same windows/scattering strategy, the same definition of ρ, the same twist and remainder thresholds, and the same decomposition of time into O(ε^{-υ}) scattering steps separated by O(ε^{-(ρ+τ)}) inner iterates, yielding the same polynomial lower bound. Minor differences are expository (e.g., the model states a uniform bound ∥DS∥ = O(ε^{-2σ}) which is consistent with the 2σ term in ρ but not singled out explicitly in the paper’s statements).

Referee report (LaTeX)

\textbf{Recommendation:} minor revisions

\textbf{Journal Tier:} specialist/solid

\textbf{Justification:}

The manuscript provides a rigorous and flexible topological-shadowing mechanism for diffusion with weak twist and partial small splitting, including explicit polynomial lower bounds for diffusion time. The construction is careful and broadly applicable. Clarity would benefit from a few targeted additions connecting the small splitting to derivative amplification and summarizing the exponent bookkeeping.