A Conservative Partially Hyperbolic Dichotomy: Hyperbolicity versus Nonhyperbolic Measures

The paper proves Theorem A: for every f ∈ PH^{1+α}_{Leb,c=1}(T^3) homotopic to a linear Anosov automorphism, either f is Anosov or it has a nonhyperbolic ergodic measure. The proof in Section 5.2 proceeds exactly along the model’s outline: (i) if f is not accessible then f is Anosov (Theorem 5.5) ; else (ii) if λ_c(Leb)=0 we are done; else λ_c(Leb)>0 implies f is mostly expanding (Theorem 3.11), whence the unique F^u-minimal set Λ either supports a nonhyperbolic measure or is uniformly hyperbolic with expanding center (Proposition 4.1) . In the latter case, semiconjugacy properties (Theorem 5.2) force the linear center of A to be expanding (Claim 5.7) ; then a transversality-vs.-foliation alternative (Theorem 5.4) yields s-transversality under accessibility, and Theorem G gives the final dichotomy . The model’s solution mirrors these steps, with one minor imprecision: it invokes a Gan–Shi criterion (Theorem 5.6) to rule out the “h maps Fu to Wu(A)” branch, but this is unnecessary because Theorem 5.4 already implies s-transversality under accessibility without assuming f is Anosov. Aside from this, the proofs coincide in structure and content.