Rigidity of the unstable foliation

Abstract

We establish a rigidity result for the unstable foliations of transitive Anosov flows on 3-manifolds: if the unstable foliations of two such flows are equivalent (that is, if there exists a homeomorphism mapping one foliation to the other), then the flows are topologically conjugate up to a constant change of time. This result partially generalizes earlier rigidity theorems for horocyclic flows on compact surfaces of negative curvature, originating in the work of Ratner. In that setting, it is known that equivalence of unstable foliations implies that the underlying surfaces are homothetic.