Generalizations of the Bonatti-Langevin example of Anosov flow and their classification up to topological equivalence

We give a complete classification of the Anosov flows on certain 3-manifolds up to topological equivalence. These manifolds are the orientable ones obtained by glueing the non-trivial circle bundle over the two-punctured projective plane along its two boundary components. We show in particular that most of these manifolds those which are not circle bundles - admit a non R-covered Anosov flow transverse to a torus. These particular Anosov flows are natural generalizations of the Bonatti-Langevin's example, and are obtained by Goodman's surgeries performed on this example. By the way, we get the first known examples of 3-manifold admitting at the same time R-covered and non R-covered Anosov flows.