Some applications of the minimal model program in arithmetic dynamics

The paper’s criterion (Theorem 7.6) that λ1(f) > 1 if and only if [f] has infinite order in π0Aut(X) for normal projective X with finitely generated nef cone is correct and is proved via a robust group-theoretic construction (D(X)) and a homomorphism Lin with finite kernel. The candidate solution reaches the same conclusion but its key converse step contains a gap: it deduces a_i ≥ 1 by asserting λ1((f^m)^{-1}) = 1 from λ1(f^m) = 1, which is not generally valid in higher dimensions. This flaw can be repaired by using det(f^{m*}) = ±1 on N^1(X)Z (so ∏ a_i = 1 and, with max a_i ≤ 1, all a_i = 1), as effectively done in the paper’s Proposition 7.5 argument.