Partially hyperbolic diffeomorphisms with one-dimensional neutral center on 3-manifolds

We prove that for any partially hyperbolic diffeomorphism having neutral center behavior on a 3-manifold, the center stable and center unstable foliations are complete; moreover, each leaf of center stable and center unstable foliations is a cylinder, a Mobius band or a plane. Further properties of the Bonatti–Parwani–Potrie type of examples of of partially hyperbolic diffeomorphisms are studied. These are obtained by composing the time \begin{document}$ m $\end{document} -map (for \begin{document}$ m>0 $\end{document} large) of a non-transitive Anosov flow \begin{document}$ \phi_t $\end{document} on an orientable 3-manifold with Dehn twists along some transverse tori, and the examples are partially hyperbolic with one-dimensional neutral center. We prove that the center foliation is given by a topological Anosov flow which is topologically equivalent to \begin{document}$ \phi_t $\end{document} . We also prove that for the original example constructed by Bonatti–Parwani–Potrie, the center stable and center unstable foliations are robustly complete.