On the Finiteness of Anosov flows on $3$-manifolds via Contact Geometry

Abstract: Following Eliashberg-Thurston and Mitsumatsu, one can associate a transverse pair of oppositely oriented contact structures to any Anosov flow. We show that the isotopy classes of these contact structures completely determine the flow up to isotopy orbit equivalence. This then implies that the number of Anosov flows modulo isotopy orbit equivalence on a closed hyperbolic $3$-manifold is finite. This approach also yields an explicit bound on the number of orbit equivalence classes of Anosov flows in terms of the number of universally tight contact structures. In addition, we show finiteness of pseudo-Anosov flows for which the complement of the singular orbits is atoroidal and, in the non-hyperbolic case, we obtain a control on the dynamics of Anosov flows on the hyperbolic pieces of the JSJ-decomposition.