Partially hyperbolic diffeomorphisms homotopic to the identity in dimension 3

Thomas Barthelme, Sergio Fenley, Steven Frankel, Rafael Potrie

We study partially hyperbolic diffeomorphisms homotopic to the identity in 3-manifolds. Under a general minimality condition, we show a dichotomy for the dynamics of the (branching) foliations in the universal cover. This allows us to give a full classifcation in certain settings: dynamically coherent partially hyperbolic diffeomorphisms on hyperbolic 3-manifolds (proving a classifcation conjecture of Hertz-Hertz-Ures in this setting), and partially hyperbolic diffeomorphisms homotopic to the identity on Seifert fibered manifolds (proving a conjecture of Pujals in this setting). In both cases, up to iterates we prove that the diffeomorphism is leaf conjugate to the time one map of an Anosov flow. Several other results of independent interest are obtained along the way.