Thermodynamic formalism for general iterated function systems with measures

The paper establishes P(ψ)=log ρ(B_q), existence of equilibrium states, and uniqueness of the X-marginal under Gâteaux differentiability with complete, self-contained arguments tailored to IFSm. It proves P≤log ρ(B_q) by exhibiting a positive eigenfunction h of B_q and testing g=h, and P≥log ρ(B_q) via a normalization/eigenfunction construction, then shows USC and compactness for existence, and uses convex subdifferential calculus for uniqueness (all within its stated hypotheses). By contrast, the model’s solution relies on an unqualified Collatz–Wielandt equality for positive operators on C(X) and on an entropy representation involving h^v−h^a that is neither the paper’s variational principle nor justified in this generality. It also mixes ψ(x,θ) potentials with the paper’s ψ(x) setting and invokes a Donsker–Varadhan inequality without checking the absolute continuity and measurability needed for J_x and h^a. Hence the model’s proof is incomplete/incorrect in key steps, while the paper’s results are correct as stated.