Entropy, pressure, ground states and calibrated sub-actions for linear dynamics

The paper’s Theorem 1 states the variational principle P_ν(A) = log(λ_A) = h_ν(μ_A) + ∫ A dμ_A under the linear-dynamics/Ruelle-operator framework and gives a proof: first an upper bound via Lemma 1, h_ν(μ) + ∫A dμ ≤ log λ_A for all T-invariant μ, and then equality is obtained at the Gibbs state μ_A by passing to the normalized cohomologous potential Ā and a simple functional inequality that, in effect, uses the Markov/concavity mechanism (implicit Jensen) to show h_ν(μ_A) + ∫A dμ_A = log λ_A (see Definition of entropy and pressure, the statement of Theorem 1, and the proof details around equations (13)–(15) ). The candidate solution proves the same result by an explicit Doob h-transform P_A(φ) = L_A(ψ_A φ)/(λ_A ψ_A), showing P_A is a Markov operator, taking μ_A fixed by P_A^*, and applying Jensen’s inequality for log to derive the reverse inequality ∫ log(L_A u/u) dμ_A ≥ log λ_A with equality at u = ψ_A. This is mathematically equivalent to the paper’s normalization argument but presented in a different, more probabilistic style. The Ruelle operator, entropy definition, and normalization step used in both accounts match the paper’s setup (Ruelle operator via integration over Ker(T) and entropy via Ruelle operator at A ≡ 0) . The paper’s proof is logically sound and complete for the stated assumptions, while the model’s proof is also correct; they differ in technique (normalization vs. explicit Doob transform/Jensen).