Anosov Flows of Codimension One

1995 The dissertation of Slobodan Simi c is approved, and is acceptable in quality and form for publication on microolm: Chair Date Date Date 1 Abstract Anosov Flows of Codimension One The main goal of this dissertation is to show the existence of global cross sections for certain classes of Anosov ows. Let be a C 2 codimension one Anosov ow on a compact Riemannian manifold M of dimension greater than three. Verjovsky conjectured that admits a global cross section and we aarm this conjecture in the following three cases: 1) if the sum, E su , of the strong stable and strong unstable bundle of is Lipschitz; 2) if E su is-HH older continuous for all < 1 and preserves volume; 3) if the center stable distribution of is of class C 1+ for all < 1 and preserves volume. We note that 1) and 3) generalize the results of Ghys from Gh3]. For showing 1), we needed to prove a natural generalization of the theorem of Frobenius on integrability of distributions which are only Lipschitz. We also show how certain transitive Anosov ows (those whose center stable distribution 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 ow 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 codi-mension one Anosov ow (when dim M > 3 and that foliation is C 1+ for all < 1), and a positive answer to a higher dimensional analog to Problem 10.4 posed by Hurder 2 and Katok in HK]. We also prove that, under an additional assumption, the lift of the synchronization of to the universal covering space of M admits a global Lyapunov function which strictly increases along the orbits and is constant on the lifts of the strong stable leaves.