Slow entropy for some Anosov-Katok diffeomorphisms

The paper explicitly proves that for C∞ AbC diffeomorphisms the upper measure-theoretic slow entropy at the polynomial scale nt vanishes (Section 6.1), via quantitative norm growth of the conjugacies and a covering-number bound imported from BKW, leading to a limit of zero for all t>0 . The model’s argument, while aiming at the same conclusion, hinges on a strong and unsubstantiated assumption that each conjugacy h_i is the identity off a set S_i with summable measures and that commutation failures are localized to S_i. This is not part of the paper’s construction—indeed, the paper uses conjugations that are not supported on summably small sets (e.g., φ_n equals the identity on half a fundamental domain, but the modified region is of fixed positive measure, not summable across stages) . Consequently, key steps in the model proof (uniform exceptional-set estimates and the Markov-inequality step yielding G_n of measure >1−ε independent of n) are not justified for the AbC diffeomorphisms under consideration. The paper’s proof is sound under its stated quantitative hypotheses; the model’s proof is incomplete under assumptions not supplied by the paper.