Hyperbolic models for transitive topological Anosov flows in dimension three

The paper proves Theorem A: any transitive topological Anosov flow on a closed 3-manifold is orbitally equivalent to a smooth Anosov flow that preserves an ergodic smooth volume form. The construction proceeds via Birkhoff sections with pseudo-Anosov return maps, an almost-Anosov atlas on the complement of finitely many periodic orbits, a Goodman-type local surgery to produce a smooth, volume-preserving Anosov model, and a criterion (Theorem B) to establish orbital equivalence; ergodicity then follows from standard Anosov theory. These steps are explicitly stated and developed in Sections 3–5, with the volume-preserving and ergodicity claims made in 5.1/5.4–5.5 (e.g., Theorem A statement and sketch, and the ergodicity remark) . The candidate solution mirrors this architecture, differing only in presentation (e.g., an explicit “coboundary” parametrization to balance Jacobians), which is consistent with the paper’s linear-affine local models and volume-preserving design.