Abstract
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.
Highlights
As in the hyperbolic (Anosov) case, invariant foliations play a substantial role in the understanding of the dynamics of partially hyperbolic systems
We remark that recently, [BBI2] have shown that absolute strong partially hyperbolic diffeomorphisms of T3 are dynamically coherent using a criterium of Brin ([Br]) which relies in this stronger version of partial hyperborbolicity in an essential way
We say that a partially hyperbolic diffeomorphism f with splitting T M = Ecs ⊕ Eu is dynamically coherent if there exists an f -invariant foliation F cs tangent to Ecs
Summary
As in the hyperbolic (Anosov) case, invariant foliations play a substantial role in the understanding of the dynamics of partially hyperbolic systems (even if many important results manage to avoid the use of the existence of such foliations, see for example [BuW2, BI, C]). We remark that recently, [BBI2] have shown that absolute strong partially hyperbolic diffeomorphisms of T3 are dynamically coherent using a criterium of Brin ([Br]) which relies in this stronger version of partial hyperborbolicity in an essential way. This has been used by Hammerlindl in [H] to obtain leaf-conjugacy results for this kind of systems. Hammerlindl we use the techniques here as well as the ones developed in [H] in order to give a classification result for strongly partially hyperbolic diffeomorphisms of T3
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