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

Abstract

<p style='text-indent:20px;'>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 Möbius band or a plane.</p><p style='text-indent:20px;'>Further properties of the Bonatti–Parwani–Potrie type of examples of of partially hyperbolic diffeomorphisms are studied. These are obtained by composing the time <inline-formula><tex-math id="M1">\begin{document}$ m $\end{document}</tex-math></inline-formula>-map (for <inline-formula><tex-math id="M2">\begin{document}$ m&gt;0 $\end{document}</tex-math></inline-formula> large) of a non-transitive Anosov flow <inline-formula><tex-math id="M3">\begin{document}$ \phi_t $\end{document}</tex-math></inline-formula> on an orientable 3-manifold with Dehn twists along some transverse tori, and the examples are partially hyperbolic with one-dimensional neutral center. We prove that the center foliation is given by a topological Anosov flow which is topologically equivalent to <inline-formula><tex-math id="M4">\begin{document}$ \phi_t $\end{document}</tex-math></inline-formula>. We also prove that for the original example constructed by Bonatti–Parwani–Potrie, the center stable and center unstable foliations are robustly complete.</p>