AN ASYMPTOTIC FOR SUMS OF LYAPUNOV EXPONENTS IN FAMILIES

The uploaded paper proves exactly the asymptotic L(f_t) = L(f_η) log|t^{-1}| + O((log|t^{-1}|)^{1−ε_N}) with ε_N = 1/(3N+4) for meromorphic families over P^1_K, by (i) deriving effective bounds for norms of homogeneous forms in t via an effective Nullstellensatz and height/Mahler–Gauss comparisons, (ii) expressing L(f_t) through the escape-rate/Green functional for the Jacobian via pushforwards, and (iii) optimizing over the iterate length to obtain the power saving; the leading coefficient is identified with the non-Archimedean Lyapunov sum using Favre’s theorem (Theorem 2 and its proof, together with Lemmas 6, 8, 9 and the final comparison with Favre’s asymptotic) . The candidate solution follows the same blueprint: lift to F_t, reduce L(f_t) to growth of pushforwards of divisors/hyperplanes using Ingram’s formula, obtain power-saving bounds via Jelonek’s effective Nullstellensatz plus height/Mahler estimates, and identify the leading coefficient by Favre. The steps, hypotheses, exponent, and identification of constants align closely with the paper’s sections “Effective estimates on norms of homogeneous forms,” “Lifts and pushforwards of divisors,” and “Variation of the escape rate” . Hence both are correct and use substantially the same proof.