CODIMENSION-ONE ANOSOV-FLOWS AND A CONJECTURE OF VERJOVSKY

Let Phi be a C-2 codimension one Anosov flow on a compact Riemannian m anifold M of dimension greater than three. Verjovsky conjectured that Phi admits a global cross-section and we affirm this conjecture when P hi, is volume preserving in the following two cases: (1) if the sum of the strong stable and strong unstable bundle of Phi is theta-Holder c ontinuous for all theta < 1; (2) if the center stable bundle of Phi is of class C1+theta for all theta < 1. We also show how certain transit ive Anosov flows (those whose center stable bundle is C-1 and transver sely orientable) can be 'synchronized', that is, reparametrized so tha t 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 cla ss of the center stable foliation of a codimension one Anosov flow (wh en dim M > 3 and that foliation is C1+theta for all theta < 1), and a positive answer to a higher-dimensional analog to Problem 10.4 posed b y Hurder and Katok in [HK].