EQUILIBRIUM STATES FOR OPEN ZOOMING MAPS

The paper establishes (i) finiteness of ergodic equilibrium states for Hölder hyperbolic potentials and for geometric potentials φ_t with t<t0, (ii) existence of a φ_t−Pf(φ_t)-conformal measure under topological exactness, and (iii) uniqueness of the equilibrium state when the map is strongly topologically transitive and backward separated. The argument proceeds via inducing schemes built from zooming/hyperbolic times, Sarig’s thermodynamic formalism on the associated countable Markov shifts, and projection using integrability of the inducing time; for conformal measures, a projection from the induced system is constructed after ensuring a technical condition (*) on the inducing scheme. All of these ingredients are present and consistent in the paper (Theorem B and supporting lemmas/propositions) . The candidate solution largely mirrors the paper’s inducing/Sarig/projection strategy and its pressure-gap reasoning, but it incorrectly asserts uniqueness under strong topological transitivity alone. The paper’s proofs require the additional backward separated hypothesis to guarantee that any two competing equilibrium states lift to the same induced scheme before appealing to uniqueness on the shift (Lemmas 7–8) . The candidate also omits the paper’s Condition (*) used to project conformal measures from the induced system to the original one (Section 8) . Hence, while most existence/finite-support conclusions align, the model’s uniqueness claim is overstated and thus incorrect as stated.