Height arguments toward the dynamical Mordell–Lang problem in arbitrary characteristic

The paper’s Theorem 1.1 proves that for a surjective endomorphism f of a projective variety X over an algebraically closed field, if every cohomological Lyapunov exponent µ_i(f) avoids Root = {a^{1/n} : a,n ∈ Z_{>0}}, then for any curve C and any point x with Of(x) = X, the intersection Of(x) ∩ C(K) is finite; see the statement and discussion around Theorem 1.1 and Remark 1.2 . The proof relies on height arguments, an Albanese reduction, and key linear-algebraic properties of cohomological Lyapunov exponents (notably µ_i(f^n) = µ_i(f)^n, Proposition 2.1(ii)) and structural steps in Proposition 3.6 and the Albanese argument in the proof of Theorem 1.1 . The model’s solution correctly notes µ_i(f^m)=µ_i(f)^m and that µ_i(f)∉Root implies µ_i(f^m)∉Z for all m, then cites the same theorem to conclude finiteness; this is logically sound and uses a condition equivalent to the paper’s hypothesis, since µ∈Root iff some power µ^m∈Z_{>0}. The only quibble is phrasing: the model states the theorem as requiring non-integer exponents for every iterate; the paper phrases it as µ_i(f)∉Root, but these are equivalent via Proposition 2.1(ii). Overall, the paper provides the substantive proof, while the model gives a correct, brief application of the paper’s main result.