SMOOTH SURFACE SYSTEMS MAY CONTAIN SMOOTH CURVES WHICH HAVE NO MEASURE OF MAXIMAL ENTROPY

The paper’s Theorem 1.8 states exactly the three properties and implications the model addresses, including compact and h‑expansive cases, and proves: (1) ⇒ (2) and (3) ⇒ (2) in general; (3) ⇒ (1) for compact Z; and, under h‑expansiveness, (2) ⇒ (1) ⇒ (3), hence full equivalence for analytic Z with positive Bowen entropy. These appear verbatim in the paper’s statements and proofs (Theorem 1.8; Theorem 7.1 for compact Z; Theorem 8.1 for h‑expansive systems, and the easy direction (1) ⇒ (2)) . The model’s arguments for (1) ⇒ (2), (3) ⇒ (2), and (3) ⇒ (1) (compact Z) align in spirit with the paper’s Carathéodory/Bowen‑ball framework and use of local entropy and Frostman‑type constructions. However, the model’s final step under h‑expansiveness informally promotes attainment of the supremum by an invariant measure via upper semicontinuity and compact approximations; this skips nontrivial measurability/compactness issues (µ(Z)=1 is not a closed constraint), and it misstates the variational principle as a supremum over invariant measures rather than over all Borel measures with lower entropy. The paper handles these subtleties carefully (e.g., Lemma 8.2, Theorem 8.1), so the conclusions match, but the model’s attainment argument is incomplete. Overall: same results, paper complete; model essentially correct on the main implications with a gap in the h‑expansive attainment step.