Codimension one Anosov flows and a conjecture of Verjovsky

Abstract

Let $ \Phi $ be a $C^2$ codimension one Anosov flow on a compact Riemannian manifold $M$ of dimension greater than three. Verjovsky conjectured that $ \Phi $ admits a global cross-section and we affirm this conjecture when $ \Phi $ is volume preserving in the following two cases: (1) if the sum of the strong stable and strong unstable bundle of $\Phi$ is $ \theta $-Hölder continuous for all $ \theta < 1 $; (2) if the center stable bundle of $ \Phi $ is of class $ C^{1 + \theta} $ for all $ \theta < 1 $. We also show how certain transitive Anosov flows (those whose center stable bundle is $C^1$ 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 $ e^t $. 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 $ \dim M > 3 $ and that foliation is $ C^{1 + \theta} $ for all $ \theta < 1 $), and a positive answer to a higher-dimensional analog to Problem 10.4 posed by Hurder and Katok in [HK].