Amie told me today about their new result on perturbation of a volume-preserving Anosov flow in three dimensions. In order to not forget what it’s about, I decided to write a sketch of what I still remember here:
So, you are given a volume preserving Anosov flow in some three-manifold (and since it’s volume preserving and Anosov and three dimensional, of course we have one dimensional stable and unstable manifolds), let be its time-1 map. Consider a perturbation of . We are interested in when is the perturbed map still a time-1 map of a flow.
Note that we know partial hyperbolicity is an open property, our perturbed map will still be a partially hyperbolic diffeo. However in general it would no longer be a time 1 map of a flow. It turns out that we can tell whether or not it’s a time-1 map just by looking at the center foliation. (some condition to do with whether some measure on the center is atomic…I can’t recall)
Furthermore this infact don’t have much to do with the fact it’s a perturbation of the Anosov flow: we may start with any volume-preserving partially hyperbolic diffeomorphism in three-manifold , assuming the diffeo preserves its center foliation (or more generally if it permutes each center leaf peroidically), then it’s time-one map of a flow precisely when their condition on the center foliation holds. Note that the center leaves are automatically preserved if the map was a perturbation of the Anosov flow.
Note that restriction our attention to volume preserving flows is essential in obtaining any of such results since in part it guarantees a dense set of periodic orbits. I’m suppose to check Franks and William’s paper on “Anomolous Anosov Flows” in which they gave many examples of different non-volume-preserving Anosov flows. The question of whether or not it’s possible to classify all Anosov flows (in the sense presented in the paper) is still open.