FIXED POINTS OF KOCH’S MAPS

The paper proves a clean trichotomy (Theorem A′) for eigenvalues μ of DG_{k,m} at fixed points, via a conjugate model F_{k,m} and a precise spectral analysis of the transpose L* on an invariant complement, culminating in Proposition 4.12: either μ=0 (k′=0), or μ is an eigenvalue of G_{k′,m′} at a fixed point outside the post‑critical set, or μ^m = λ^{m/m′} with μ^{m′}≠λ (all mutually exclusive) . The candidate solution reaches the same classification but relies on a flawed step: it assumes a one‑dimensional space of m′‑periodic particular solutions when μ^{m′}≠λ and “rescales” such a solution to force a closing condition. In fact, in that non‑resonant case the m′‑periodic particular solution is unique and cannot be rescaled, so the key equation used to deduce μ^m=λ^{m/m′} is not justified. The paper’s argument—based on the cyclic action of L* on a natural basis and a direct eigenvalue computation—does not suffer from this gap and is correct .