Partially hyperbolic diffeormorphisms, ergodicity, and transverse foliations in dimension 3
We give a complete topological classification of (chain-)transitive partially hyperbolic diffeomorphisms in 3-manifolds in terms of Anosov flows, completing a program proposed by Pujals. In particular, this also allows to give a full answer to the ergodicity conjecture of Hertz-Hertz-Ures for partially hyperbolic diffeomorphisms in dimension 3. This is achieved by showing a result about pairs of transverse $2$-dimensional foliations in 3-manifolds with Gromov hyperbolic leaves (Theorem B), which may be of independent interest. This paper will be superseded by the forthcoming article together with Andy Hammerlindl, where we obtain a much more general version of Theorem B of this paper. This will allow to remove the assumption of chain transitivity of the partially hyperbolic diffeomorphism and will provide a full classification of partially hyperbolic diffeomorphisms in dimension $3$. For this reason, this paper will remain a permanent preprint.