UNIQUENESS OF EQUILIBRIUM MEASURE FOR A FAMILY OF PARTIALLY HYPERBOLIC HORSESHOES

The paper introduces projective hyperbolic potentials (Definition 2.2) and proves Theorem C: uniqueness of the equilibrium state for (G, ϕ=φ|Ω), hence for the horseshoe F by Theorem B (equivalence via the semiconjugacy π with π∘F^{-1}=G∘π). The proof route uses Proposition 7.1 to show ϕ is expanding/hyperbolic and then invokes a general uniqueness theorem for hyperbolic Hölder potentials on non-uniformly expanding local homeomorphisms (Ramos–Viana) to conclude uniqueness; Theorem B transfers this back to F . The candidate solution proves the same conclusion by a different, standard route: build a Climenhaga–Thompson decomposition using hyperbolic times to get a good collection with Bowen property and non-uniform specification, use the pressure gap provided by (D2), deduce uniqueness for (G,ϕ), and then transfer uniqueness to (F,φ) via the semiconjugacy and a relative variational principle. This aligns with the structure and assumptions in the paper (Sections 5–7) and does not contradict it, though it relies on a different toolkit. Minor clarifications noted below do not affect the main conclusion.