Codimension one Anosov flows and a conjecture of Verjovsky

Let Φ be a C² codimension one Anosov flow on a compact Riemannian manifold M of dimension greater than three. Verjovsky conjectured that Φ admits a global cross-section and we affirm this conjecture when Φ is volume preserving in the following two cases: (1) if the sum of the strong stable and strong unstable bundle of Φ is θ-Hölder continuous for all θ < 1; (2) if the center stable bundle of Φ is of class C1 + θ for all θ < 1. We also show how certain transitive Anosov flows (those whose center stable bundle is C¹ and transversely orientable) can be ‘synchronized’, that is, reparametrized so that the strong unstable determinant of the time t map (for all t) of the synchronized flow is identically equal to eᵗ. Several applications of this method are given, including vanishing of the Godbillon–Vey class of the center stable foliation of a codimension one Anosov flow (when M > 3 and that foliation is C1 + θ for all θ < 1), and a positive answer to a higher-dimensional analog to Problem 10.4 posed by Hurder and Katok in [HK].