Arithmetic Degrees are Cohomological Lyapunov Multipliers

The paper proves precisely the statement the model labeled as likely open: for a surjective endomorphism f of a normal projective variety X and a point x with Zariski-dense orbit, the arithmetic degree satisfies α_f(x) ∈ {µ_i(f)} ∩ R_{≥1} (Theorem 1.2) . The proof works over finitely generated fields using Moriwaki heights, and it leverages: (i) functoriality through the Albanese to obtain α_f(x) ≥ λ_1(g) for the induced endomorphism g on Alb(X) and to dispatch the case α_f(x)=λ_1(g) via the inclusion of cohomological Lyapunov multipliers under semi-conjugacy ; (ii) a lifting lemma that splits off N^1(X)_R-eigen-directions into Pic(X)_R above the spectral radius on Pic^0(X) ; (iii) canonical height vectors for Jordan blocks to control growth from the big part of N^1(X)_R , ; (iv) an explicit polynomial–exponential growth bound for lower-rate components (Corollary 3.3) to contradict any α strictly between adjacent µ_i’s . The paper also recalls the monotonicity and functoriality of µ_i(f) and the standard upper bound α_f(x) ≤ λ_1(f) in this setting, along with the Moriwaki-height normalization α_f(x) ≥ 1, tying all ingredients together , , . In short, the paper establishes the general membership result; the model’s claim that no such proof is known is outdated relative to this paper.