Orbit equivalence of pseudo-Anosov flows in 3-manifolds

Resumen: We will talk about the result that given two pseudo-Anosov flows in a 3-manifold, at least one of which is transitive, and so that they have the same spectrum, and at least one of them does not have a tree of scalloped regions, then the flows are (isotopically) orbitally equivalent. The spectrum is the set of conjugacy classes in the fundamental group, which are represented by periodic orbits of the corresponding flow. There is a refined statement that takes into account the existence of trees of scalloped regions, where one still gets an orbitally equivalence result. The proof is done by analyzing the actions on the corresponding orbit spaces, each of which is a bifoliated plane. Before talking about the proof, we give some counterexamples when there are trees of scalloped regions.