On Gibbs measures for almost additive sequences associated to some relative pressure functions

The paper proves Theorem 3.1 under Setting (A) by bounding lim inf/lim sup ratios and invoking Lemma 3.2 to obtain e^{-C} ≤ f_k/e^{S_k f̂} ≤ e^C, which yields (1) and (2), and then transfers Gibbs/equilibrium properties (and uniqueness when F has bounded variation) via Lemma 3.1; it likewise treats the weakly almost additive case and Setting (B) (Theorem 3.2) . The model solves the same core problem by a telescoping identity: defining g_n = F_n − F_{n−1}∘σ and f̂ = lim g_n, then showing S_n f̂ = lim_m [F_{n+m} − F_m∘σ^n] and using almost/weakly almost additivity to obtain |F_n − S_n f̂| ≤ C (or C_n), hence (1)–(2), and transferring the Gibbs/equilibrium conclusions. The proofs are logically consistent, reach the same conclusions, and differ mainly in technique (lim inf/lim sup versus telescoping). A small gap in the model’s uniqueness-of-Gibbs-for f̂ step (it appeals to “Gibbs ⇒ equilibrium” for a Borel potential without justification) can be patched exactly as in the paper by using the comparability e^{−C} ≤ f_n/e^{S_n f̂} ≤ e^C and Lemma 3.1(ii) to pull back any Gibbs measure for f̂ to a Gibbs measure for F, restoring uniqueness .