UNIQUENESS AND STABILITY OF EQUILIBRIUM STATES FOR RANDOM NON-UNIFORMLY EXPANDING MAPS.

The paper establishes, under hypotheses (I)–(VI) and small-variation Hölder potentials, existence of a random Ruelle–Perron–Frobenius triplet (h_w, ν_w, λ_w), builds the invariant measure μ_w = h_w ν_w, proves the pressure identity PF|θ(φ) = ∫ log λ_w dP(w), uniqueness among non-uniformly expanding measures, and stability of both the equilibrium state and pressure (Theorems A–D). These points are explicit in the setup and main results and in the proofs via cone techniques, Jacobian/entropy formulas, Jensen-based uniqueness, and a direct compactness-and-convergence argument for stability . The candidate solution mirrors this program: cone contraction on Hölder cones; construction of the random RPF data and invariant μ; pressure identification; variational principle; non-uniform expansion using the (VI) frequency bound; and stability under perturbations. Its main methodological differences are (a) it asserts a full Bowen-ball Gibbs property (the paper proves a Gibbs-type estimate on hyperbolic times rather than uniformly in n,x), and (b) it invokes Keller–Liverani for stability whereas the paper uses a direct weak-* subsequence plus operator-convergence approach. With these caveats noted, the core conclusions and logical structure agree, so both are correct, with the model’s Step 3 requiring the usual restriction to hyperbolic times (or additional distortion hypotheses) to match the paper’s scope .