Rational Maps with Rational Multipliers

The paper proves the exact statement (Theorem 7) and its argument is complete: reduce to a model over K via the isospectral locus (Silverman + McMullen), build Galois-stable unions of cycles, apply Autissier-type equidistribution, use a convex truncation max(log||f′||, m) to deduce χ ≤ L(f), then combine a periodic-approximation lemma with Zdunik’s rigidity to obtain a contradiction unless f is power/Chebyshev/Lattès . The model follows the same blueprint but makes a crucial Jensen step with φR = min(log||f′||, R) in the wrong inequality direction; this prevents deriving the needed bound limsup χ ≤ L(f). With the standard fix (truncate using max rather than min), the model’s approach would match the paper’s proof.