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

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  • 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.

    Further properties of the Bonatti–Parwani–Potrie type of examples of of partially hyperbolic diffeomorphisms are studied. These are obtained by composing the time $ m $-map (for $ m>0 $ large) of a non-transitive Anosov flow $ \phi_t $ 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 $ \phi_t $. We also prove that for the original example constructed by Bonatti–Parwani–Potrie, the center stable and center unstable foliations are robustly complete.

    Mathematics Subject Classification: Primary: 37D30; Secondary: 37C05, 57R30.

    Citation:

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  • Figure 1.  The intersection between the local stable manifold of $ f^{n_i}(L) $ and the local unstable manifold of $ \mathscr{F}^c(w_1) $

    Figure 2.  Defining $ h^s $ on a center stable leaf restricted to $ \tilde{M}^-\backslash\tilde{\mathscr{A}} $

    Figure 3.  The real lines and the dashed lines denote the leaves of the lifts of foliations $ {\mathscr{F}}^s_i $ and $ {\mathscr{F}}^u_i $ to the universal cover, respectively

    Figure 4.  The real lines and the dashed lines denote the leaves of the lifts of foliations $ {\mathscr{F}}^u_i $ and $ {\mathscr{F}}^s_i $ to the universal cover, respectively

    Figure 5.  The dashed line denotes the center leaf obtained by the intersection of center stable and center unstable leaves

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