Equilibrium measures of the natural extension of β-shifts

The paper’s Theorem 2.1 proves exactly the equivalence stated in the candidate solution: for functions of bounded total oscillations on the natural extension Σβ, sublinear zβ(n) yields that every equilibrium state is weak Gibbs, whereas positive limsup of zβ(n)/n precludes weak Gibbs. The proof uses window sets En with bijective projections Jr−n,ns, partition sums Ξn, uniform distortion for BTO potentials (Lemma 2.4), and two comparison estimates for partition functions (Lemmas 2.5–2.6), culminating in cylinder bounds for any equilibrium measure via convexity/tangent-functional arguments and then the two cases in §3.2. All of these ingredients match the model’s outline. Minor discrepancies are cosmetic: the model’s ‘b-to-1’ counting is more precisely bounded in the paper by zβ(c1)+2; and the model phrases the comparison via a gluing construction producing Ξn(ψ)^2–type bounds, while the paper works directly with ratios Ξn[rk,ℓ](v)/Ξn[rk,ℓ](u). These do not affect correctness. See the statement of Theorem 2.1 and its proof outline, the En/pressure definitions, the BTO distortion lemma, and the partition function estimates in Lemmas 2.5–2.7 and Lemmas 3.1–3.2, and the negative case in §3.2 (Equations (3.22)–(3.24)) .