DYNAMICAL DEGREES, ARITHMETIC DEGREES, AND CANONICAL HEIGHTS: HISTORY, CONJECTURES, AND FUTURE DIRECTIONS

The paper proves the same statement (Proposition 56) by blowing up X along Z and applying Vojta on the blow-up, then using the “no accumulating subvarieties” hypothesis to ensure the orbit eventually avoids the exceptional closed set, yielding h_{X,Z}(f^n(P)) ≤ ε h_{X,H}(f^n(P)) for all large n and hence the ratio tends to 0. The candidate solution follows the same blueprint, but with more care: it uses the general blow-up formula K_{X̂} = π^*K_X + (codim Z − 1)·E, tracks O(1) terms, compares heights on X̂ and X, and (optionally) invokes dynamical height growth to kill additive constants. Minor slips in the paper’s proof (missing the factor codim Z − 1 and implicitly identifying an ample height on X̂ with the pullback height from X) are easily repaired and do not affect the conclusion. Overall, both are correct, with substantially the same proof strategy via blow-up + Vojta. Key places in the paper include the definitions and question setting (Definition 51–Question 53) and the statement/proof of Proposition 56, where the argument appears explicitly.