PRESSURE INEQUALITIES FOR GIBBS MEASURES OF COUNTABLE MARKOV SHIFTS

The paper states and proves Theorem 1.1: for a topologically mixing countable Markov shift with a Hölder potential and a Gibbs measure m, there exists a constant a>0 such that for any T-invariant µ and f in the Lipschitz space L, |m(f)−µ(f)| ≤ a‖f‖_L (PGS(φ) − P_µ(φ))^{1/2}. The statement appears explicitly and the proof is given via Sarig’s GRPF theory, a spectral-gap estimate for the transfer operator, a telescoping in n of T̂^n f, Pinsker’s inequality, and the identity µ(D_p(q)) = PGS(φ) − P_µ(φ) derived from the information function formula I_m = −[φ + log h − log h∘T − PGS(φ)] (equation (4)), all provided in the text. See the theorem statement and proof components, including the spectral-gap inequality (3), the transfer-operator telescoping, Pinsker-bound step (5), and reduction of µ(D_p(q)) to the pressure gap, in the paper’s Sections 1–2 . By contrast, the candidate solution’s Step 4 makes a crucial, incorrect oscillation bound: it asserts that for the martingale-difference decomposition D_k = E_m[f|ℱ_k] − E_m[f|ℱ_{k+1}], one has ‖D_k‖_∞ ≤ var_{k+1} f. This confuses tail σ-algebras with initial cylinders. For instance, for f depending only on x_0, var_{k+1} f = 0 for all k, yet D_k generally need not vanish (conditioning on deeper tail information can change E_m[f|ℱ_k]), contradicting the claimed bound. The rest of the model’s argument relies on this bound and thus collapses. The paper’s proof avoids this pitfall by controlling fn − m(f) via spectral gap and summability in n, not via var_{k+1} f, and is therefore correct and complete in its stated setting .