Constructing equilibrium states for Smale spaces

The paper proves Theorem 3.5 by (i) tilting a conditional Gibbs measure on an unstable plaque, (ii) establishing the growth rate of the normalizing integral via a plaque growth lemma (Lemma 4.1) showing lim (1/n) log ∫_{W^u} e^{S_n(G2−G1)} dμ^u_{G1} = P(G2)−P(G1), and (iii) combining an entropy estimate (Misiurewicz-type) with the variational principle to conclude that any weak* limit is an equilibrium state for G2; uniqueness for Hölder G2 then gives convergence of the entire sequence. The candidate solution follows the same construction but identifies the plaque growth rate with P(G2) using a symbolic coding argument for Smale spaces, and then derives an upper Gibbs-type inequality on global Bowen balls to reach the variational equality. Both arguments are logically sound and reach the same conclusion; the proofs differ mainly in how they prove the plaque growth rate and how they turn plaque estimates into the variational equality. The paper’s argument avoids symbolic coding and uses a direct geometric/separated–spanning set method, whereas the model leverages coding to a mixing SFT. Both are correct.