On entropy, entropy-like quantities, and applications

The paper defines H2(T) and CE(T,1) (Definition 24) and states Theorem 25: under the hypotheses (compact metric Ω, continuous T, non-atomic invariant µ) one has H2(T) ≤ CE(T,1) = hµ(T), noting the H2 ≤ CE step follows from Jensen’s inequality . It also recalls the Brin–Katok local entropy formula (Theorem 22), which underpins the identification with Kolmogorov–Sinai entropy . The candidate solution mirrors this: (i) Jensen for H2 ≤ CE; (ii) CE ≤ hµ via small-mesh partitions and αk-cylinders contained in Bowen balls; (iii) CE ≥ hµ via the Brin–Katok local entropy formula plus Fatou. The only minor quibble is that the candidate informally attributes the existence of the inner k→∞ limit in CE(T,1) to Step 3; the paper instead cites Verbitskiy for this existence (Lemma 2.14 in [88]) . Aside from this small precision, both are correct and follow the same conceptual route.