EQUILIBRIUM STATES FOR MAPS ISOTOPIC TO ANOSOV

The paper proves the stated claims (Theorem 4) using Ledrappier–Walters with property (H) to reduce pressure/entropy to the factor, then a careful geometric-measure argument: when μ(C)=0, uniqueness follows; when μ(C)=1, the authors build fiberwise partitions P and Q, use a Carathéodory-measurable parametrization along oriented 1D subfoliations to force atomic disintegration, and via a quotient argument get one atom per rectangle; with extra hypotheses they obtain virtual hyperbolicity and non-uniqueness of equilibrium states (Theorem 4; see also Theorem 3 and Corollary 2, and the proofs in Sections 4 and 6) . By contrast, the model’s proof hinges on unjustified steps in relative entropy: it asserts h_μ(f|σ(H))=0 implies H_μ(α|σ(H))=0 for a fiber-adapted finite partition α and then that conditional measures are supported on “corner-orbits.” These implications are neither standard nor established in the write-up and conflict with the paper’s more careful construction; the model also asserts existence of invariant measurable sections without the measurable-selection work present in the paper (Remark 4/Lemma 1 for the factor reduction; Section 6 for measurability of extremal selections) .