Dynamical mechanism behind ghosts unveiled in a map complexification

The paper establishes, first for the concrete ecological map F_ε and then for a general real‑analytic family H_ε, that the post–saddle‑node passage time equals 2π divided by the imaginary part of the repelling complex multiplier, yielding the inverse square‑root scaling N_ε ≈ 2π/Im λ + K and, concretely, N_ε ≈ π/(μ x_c^{3/2} ε^{1/2}) + K for n=1 (Theorem II.1 and Theorem II.3). It derives this via a contour integral/holomorphic index argument and an approximation of the iterate count by an integral across the bottleneck . The candidate solution proves the same conclusions with a slightly different and more explicit route: (i) Rouché/implicit-function analysis to locate the complex fixed points and expand their multipliers (matching Proposition II.2) , (ii) a Fatou–Abel coordinate to justify that the discrete iterate count equals a principal value integral plus a bounded constant, and (iii) a residue computation of that principal value giving 2π/Im(λ) up to an ε‑independent constant. Minor presentational differences aside (e.g., the paper writes N_ε,δ ≈ ∫ dx/(F_ε−x) and introduces a curve-dependent correction I_δ^ε, while the model uses PV integrals and emphasizes O(1) constants), the methods are consistent and yield the same formulas and asymptotics, including the general H_ε case with the n<2m, n odd hypotheses .