Pseudo-Anosov Reeb flows and contact structures

Pseudo-Anosov flows, originally introduced by Thurston, are a broad class of hyperbolic flows on 3-manifolds that are Anosov except along a finite link of singular closed orbits. A longstanding conjecture states that any 3-manifold carries only finitely many transitive pseudo-Anosov flows, up to the appropriate type of equivalence. In this talk, I will explain a proof of this conjecture for a large class of pseudo-Anosov flows that are Reeb, in the appropriate sense. This extends recent work by Barthelme-Bowedn-Mann for Reeb Anosov flows. We also address a number of questions of Barthelme. The proof uses a novel flavor of contact homology. This is joint work with Yijie Pan (USC).