Non-wandering partially hyperbolic diffeomorphisms with quasi-isometric center on closed 3-manifolds are either skew products over a torus Anosov map or discretized Anosov flows, and volume-preserving ones are ergodic.
Partially hyperbolic diffeomorphisms homotopic to the identity in dimension 3, Part I: The dynamically coherent case
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abstract
We study 3-dimensional dynamically coherent partially hyperbolic diffeomorphisms that are homotopic to the identity, focusing on the transverse geometry and topology of the center stable and center unstable foliations, and the dynamics within their leaves. We find a structural dichotomy for these foliations, which we use to show that every such diffeomorphism on a hyperbolic or Seifert fibered 3-manifold is leaf conjugate to the time one map of a (topological) Anosov flow. This proves a classification conjecture of Hertz-Hertz-Ures in hyperbolic 3-manifolds and in the homotopy class of the identity of Seifert manifolds.
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Partially Hyperbolic Dynamics with Quasi-isometric Center
Non-wandering partially hyperbolic diffeomorphisms with quasi-isometric center on closed 3-manifolds are either skew products over a torus Anosov map or discretized Anosov flows, and volume-preserving ones are ergodic.