AN IMPROVED CLIMENHAGA-THOMPSON CRITERION FOR LOCALLY MAXIMAL SETS

The paper’s Theorem A assumes (I0)–(III) and a strict pressure gap for obstructions to expansivity and proves there is a unique equilibrium state for ϕ|Λ, supported on Λ and ergodic. The proof works directly at the flow level by (i) showing P(O(U)) = P(O(U1)) = P(Λ×R+) = P(ϕ;Λ), (ii) constructing µ via weighted separated sets in O(U1) and time-averaging, and (iii) establishing lower and upper Gibbs bounds (on G0 and Λ, respectively) to conclude uniqueness and ergodicity (Lemma 3.1; Lemma 4.1; Proposition 5.5; Lemma 4.2) . By contrast, the model’s argument reduces to the time–1 map and invokes the Climenhaga–Thompson uniqueness theorem for homeomorphisms; its key step claims that the flow-level pressure gap for obstructions to expansivity implies the same gap for the time–1 map. This is incorrect: since Γε^flow(x) ⊂ Γε^map(x), we have NE_flow(ε;Λ) ⊂ NE_map(ε;Λ) (cf. the paper’s definition of obstructions to expansivity) and thus the obstruction pressure for the map is ≥ (not ≤) the obstruction pressure for the flow; the model’s inequality is reversed, so the required map-level pressure gap is not justified . The paper’s flow-level Gibbs method avoids this pitfall and appears complete and correct under its stated hypotheses.