Symplectic Rigidity of Anosov Flows Under Orbit Equivalence

A classical construction of Mitsumatsu, later generalized by Hozoori, associates to any (oriented) Anosov flow on a closed 3-manifold M a Liouville structure on the thickening [−1,1]×M. This construction also extends to suitable taut foliations.

From the dynamical viewpoint, it is natural to study Anosov flows up to orbit equivalence, i.e., homeomorphisms sending (unparametrized) orbits to orbits. Such maps are typically not smooth, so the effect on the associated Liouville structures is far from obvious. In this talk, I will present joint work with Jonathan Bowden showing that orbit equivalent Anosov flows induce exact symplectomorphic Liouville structures. A similar result holds for taut foliations. Consequently, the symplectic invariants (Floer homology, Fukaya category, etc.) attached to Anosov flows are preserved under orbit equivalence.