Periodic Points and Arithmetic Degrees of Certain Birational Self-Maps

The paper proves the bounded-height result by constructing, on a common resolution, a big divisor D = φ*H + ψ*H − 3π*H and deriving a two-sided height inequality h(f^n(R)) + h(f^{-n}(R)) ≥ A h(R) − C on a Zariski open set; Lemma 3.1 then yields bounded height for periodic points avoiding a proper closed Z. The construction uses Siu’s numerical criterion and Dang’s inequality to ensure bigness and thereby a uniform lower bound for the height associated to D off its base locus. This argument is complete and correct. By contrast, the model proposes a stronger one-sided inequality h_H(f^m(Q)) ≥ a_m h_H(Q) − C by asserting an unsupported decomposition ψ*H ≡ a_m π*H + E_m + N_m with E_m effective, which is generally false (pseudo-effectivity does not ensure an exact effective representative of a numerical class). Without establishing bigness and an appropriate lower bound off an augmented base locus, the claimed inequality and the resulting conclusion do not follow.