ENDOMORPHISMS OF QUASI-PROJECTIVE VARIETIES - TOWARDS ZARISKI DENSE ORBIT AND KAWAGUCHI-SILVERMAN CONJECTURES

The paper proves all three cases of ZDO in Theorem 1.12: (1) surfaces with d1>1 and d1≥d2, (2) threefold automorphisms with d3=1<d1, and (3) the cohomologically hyperbolic regime d1>max_{i≥2}di for smooth X, by introducing the Small Dynamical Degree (SDD) condition and showing ZDO holds under SDD (Theorem 8.4), then verifying SDD in cases (1)–(2) (Proposition 8.5) and, crucially, in case (3) (Proposition 8.6). This yields Theorem 1.12 asserting ZDO for all three settings . By contrast, the candidate solution (i) treats (2) only under an extra “simple cohomology” hypothesis (unnecessary here) and (ii) classifies (3) as likely open as of 2021, which is contradicted by the paper’s unconditional proof (over Q) via arithmetic-degree methods. The paper also records the standard functorialities about Zariski-dense orbits and generically finite maps used in the candidate’s reductions (Remark 3.5 and Proposition 3.13) . Therefore, the model’s assessment of (3) is incorrect relative to this April 2021 paper.