Anosov flows and Anosov diffeomorphisms are the archetypical examples of a uniformly hyperbolic (a.k.a. chaotic) dynamical systems, and, as such, have been widely studied since their introduction by D. Anosov in the 60's. In the past 20-30 years, a lot of work has been on the topological study of Anosov flows in dimension 3.
There are two main takeaways from these works: First, there is an interesting, and not fully understood, relationship between the topology of the manifold and the dynamics of the flows that live on it. Second there is a huge wealth of example of Anosov flows in 3 manifolds exhibiting many different types of properties.
For Anosov flows in higher dimensions on the other hand, almost nothing is known. In particular, almost no examples are known: Aside from the algebraic ones, Franks and Williams suggested a construction in 1979, but nothing came up since then.
In this talk, I will present a partial classification result about certain types of Anosov flows, which shows in particular that Franks and Williams suggestion does not work. If time permits I will explain how one can modify their construction in order to get true examples in higher dimensions.
This is joint work with C. Bonatti, A. Gogolev et F. Rodriguez Hertz.