On hyperbolic and Seifert fibered 3-manifolds, dynamically coherent partially hyperbolic diffeomorphisms homotopic to the identity are, up to iterate, leaf conjugate to time-one maps of topological Anosov flows.
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Partially hyperbolic diffeomorphisms homotopic to the identity in dimension 3, Part I: The dynamically coherent case
On hyperbolic and Seifert fibered 3-manifolds, dynamically coherent partially hyperbolic diffeomorphisms homotopic to the identity are, up to iterate, leaf conjugate to time-one maps of topological Anosov flows.