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Partial hyperbolicity and foliations in $\mathbb{T}^3$

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  • We prove that dynamical coherence is an open and closed property in the space of partially hyperbolic diffeomorphisms of $\mathbb{T}^3$ isotopic to Anosov. Moreover, we prove that strong partially hyperbolic diffeomorphisms of $\mathbb{T}^3$ are either dynamically coherent or have an invariant two-dimensional torus which is either contracting or repelling. We develop for this end some general results on codimension one foliations which may be of independent interest.
    Mathematics Subject Classification: Primary: 37C05, 37C20; Secondary: 37C25, 37C29, 37D30, 57R30.


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