UNIQUENESS OF u-GIBBS MEASURES FOR HYPERBOLIC SKEW PRODUCTS ON T4

The uploaded paper proves Theorem A: for r ≥ 3 there exists a Cr-dense and C2-open subset U ⊂ Ar such that, for every f ∈ U, there is a unique u-Gibbs measure and it coincides with the unique SRB measure. This is stated explicitly and proved using (i) Obata’s trichotomy for u-Gibbs measures (Theorem 2.3), (ii) a Cr perturbation (Proposition 3.1) that breaks the infinitesimal holonomy rigidity, (iii) accessibility to rule out Ess ⊕ Euu-tori, and (iv) Bowen’s uniqueness of SRB for transitive Anosov systems. See the statement of Theorem A and its proof sketch in the paper, including the use of accessibility and Proposition 3.1, and the reference to Bowen for SRB uniqueness . The trichotomy used is precisely Theorem 2.3 in the paper (SRB vs. infinitesimal rigidity vs. finitely many tori tangent to Ess ⊕ Euu) . The background setting Ar and regularity/holonomy continuity assumptions are spelled out in Section 2 (center-bunching, pinching, and holonomies) . The candidate solution mirrors the paper’s proof essentially step-by-step, citing the same ingredients (HS17 accessibility, Ob21 trichotomy, perturbation to break holonomy rigidity, and Bowen for SRB uniqueness). The only minor omission in the candidate is not stating the r ≥ 3 requirement upfront (needed for Theorem 2.3 and Theorem A), but it is implicit in their references to the 2022 result. Hence, both are correct and follow substantially the same argument.