Existence and Finiteness of equilibrium states for some Partially hyperbolic endomorphisms

The uploaded paper proves existence of equilibrium states for partially hyperbolic endomorphisms with a dominated splitting into one-dimensional center subbundles by passing to the natural extension, proving h-expansiveness via fake foliations and a Pliss-type argument so that each bi-infinite Bowen ball lies in a single 1D center curve, hence has zero local entropy; then Misiurewicz yields upper semicontinuity of entropy and the maximizing measure projects downstairs. This is exactly the structure in the candidate solution. The paper’s Proposition 4.5 asserts Bowen balls are contained in a center curve, leading to h-expansiveness and existence of equilibrium states (proof of Theorem B) . The natural-extension measure correspondence and entropy equality are stated explicitly (Propositions 2.5–2.6) , and the asymptotic/h-expansive to USC step is noted (Theorem 2.3 and Remark 2.4) . Minor presentational gaps exist (e.g., the descent of USC to the base space is asserted tersely), but the argument is standard and correct. The model’s solution mirrors these steps, including the restriction to compact invariant subsets, which the paper also claims in Theorem B’s statement .