Rigidity of the unstable foliation

TL;DR

This paper proves a rigidity theorem for unstable foliations of transitive Anosov flows on 3-manifolds, showing that foliation equivalence implies the flows are topologically conjugate up to time change, extending previous results in negative curvature settings.

Contribution

It establishes a new rigidity result linking unstable foliation equivalence to flow conjugacy for 3-manifold Anosov flows, generalizing earlier horocyclic flow theorems.

Findings

Unstable foliation equivalence implies flow conjugacy up to time change.

Extends rigidity results from negative curvature surfaces to 3-manifolds.

Provides a partial generalization of Ratner's theorems.

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.