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Partially hyperbolic diffeomorphisms with compact center foliations

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  • Let $f\colon M\to M$ be a partially hyperbolic diffeomorphism such that all of its center leaves are compact. We prove that Sullivan's example of a circle foliation that has arbitrary long leaves cannot be the center foliation of $f$. This is proved by thorough study of the accessible boundaries of the center-stable and the center-unstable leaves.
        Also we show that a finite cover of $f$ fibers over an Anosov toral automorphism if one of the following conditions is met:
    1. 1. the center foliation of $f$ has codimension 2, or
    2. 2. the center leaves of $f$ are simply connected leaves and the unstable foliation of $f$ is one-dimensional.
    Mathematics Subject Classification: Primary: 37D30, 57R30; Secondary: 37D20.


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