Partially hyperbolic diffeomorphisms and Lagrangean contact structures

In this paper, we classify the three-dimensional contact partially hyperbolic diffeomorphisms whose stable, unstable and central distributions are smooth, and whose non-wandering set equals the whole manifold. We prove that up to a finite quotient or a finite power, they are smoothly conjugated either to a time-map of an algebraic contact-Anosov flow, or to an affine partially hyperbolic automorphism of a nil-$\mathrm{Heis}(3)$-manifold. The rigid geometric structure induced by the three dynamical distributions plays a fundamental role in the proof.