Smoothening pseudo-Anosov flows

Question

A topological Anosov flow on a closed 3-manifold can be replaced by a smooth Anosov flow using an argument of Fried: use Markov partitions to find a surface of section, put in other terms, one can blow up some closed orbits so that the flow is a suspension of a pseudo-Anosov map on a surface with boundary. Then take a smooth representative of this pseudo-Anosov map within its isotopy class, and blow down the orbits to get back the original manifold.

Is this statement (or some analogue of it) still true for pseudo-Anosov flows? The above argument doesn't work anymore since the singular orbits cannot lie in the interior of a Markov rectangle, so one cannot find a surface of section near these singular orbits in the same way. Is an alternate way of thinking about all of this that generalizes more easily to pseudo-Anosov flows? Any help is appreciated!

Answer 1

In fact, the argument by Fried that you describe is incomplete. It is unclear that such a scheme can be made to work and there is evidence that it may not work (why should the blow down be smooth, and if it were, why would it be a smooth Anosov flow rather than a smooth topological Anosov flow).

However, the result has been clarified and rigourously proved recently in Mario Shannon's thesis http://www.theses.fr/2020UBFCK035#

It is likely that his techniques also respond to your problem and are the technique you seem to be looking for.