Non-algebraic Anosov flows in high dimensions

Anosov flows and Anosov diffeomorphisms are the archetypical examples of uniformly hyperbolic dynamical systems, and, as such, have been widely studied since their introduction by D. Anosov in the 60s. There are many examples of Anosov flows on 3-manifolds exhibiting many different types of properties. The reason is the existence of 2 constructions process, by surgeries:

• the first process has been initiated by Handel and Thurston and generalized by Goodman and Fried: given any Anosov flow on a 3-manifolds, one can build infinitely many of them by surgeries along periodic orbit.
• the second, started with Franks and Williams “anomalous Anosov flows” in 1980, and then by myself with Langevin in 1994 and generalized with Beguin and Yu in 2017, allows us to build Anosov flows by gluing hyperbolic plugs.

In higher dimensions very few is known, due to a lack of examples. Indeed Franks and Williams announced in 1980 that their constructions holds in any dimensions $>3$ but the argument was not precisely presented. We (with T. Barthelmé, A. Gogolev, and F. Rodriguez Hertz) recently noticed that indeed their argument could not work in dimensions $>3$, but we could build examples following the same spirit.
I will try to present the transitive and non-transitive examples in [BBGH 2021] which are therefore the first examples of non-algebraic Anosov flows in dimension $>3$.