Technical Developments of DA on T3

The paper proves (with explicit constants and cone/area estimates) the existence of a C∞ derived-from-Anosov diffeomorphism f on T^3 that is partially volume-expanding with an index-2 fixed point, and shows robustness; it also proves that f^{-1} has a mostly expanding center, robust among C^{1+} maps. See Theorem 2.1 and Theorem 2.2, together with the construction and estimates in Section 3 (especially Lemma 3.2, Lemma 3.3, Proposition 3.6) and Section 4’s control criterion for Gibbs u-states . By contrast, the model’s “Claim 1” assumes that, in the 1–1–1 splitting, the minimal area expansion over planes containing E^uu is attained on E^uu ⊕ F^ss with value |λ_u||λ_s|. That is generally false unless the eigenbasis is orthonormal (equivalently, the linear map is normal with respect to the chosen metric). For the model’s chosen non-symmetric integer matrix A, this claim fails: the minimal 2-plane expansion over planes containing E^uu can be < 1 even though |λ_u||λ_s| > 1, so A need not be partially volume-expanding in the Euclidean metric. The paper avoids this pitfall by choosing a symmetric base automorphism (orthogonal eigendirections) and then carrying out careful wedge-product lower bounds under the local modification. The model also asserts “mostly contracting center” for f without quantitative control of return frequencies to the modified region, whereas the paper provides the necessary measure-theoretic control to conclude positivity of center exponents for all Gibbs u-states of f^{-1} and the robustness statement .