Rigidity of pseudo-Anosov flows transverse to R-covered foliations

Pseudo-Anosov flows are extremely common in three manifolds
and they are very useful. How many pseudo-Anosov flows are there in a manifold up to topological conjugacy? We analyse this question in the context of flows transverse to a given foliation F. We prove that if F is R-covered (leaf space in the universal cover is the real numbers) then there are at most two pseudo-Anosov flows transverse to F. Usually there is at most one, but if there are two, then the foliation F blows down to a foliation topologically conjugate to the stable foliation of a particular type of an Anosov flow. The results use the topological theory of pseudo-Anosov flows and the universal cover for foliations.