Partial hyperbolicity on 3-dimensional nilmanifolds (Q379521)

A diffeomorphism is partially hyperbolic if the tangent bundle splits into three subbundles \(TM=E^{s}\oplus E^{c}\oplus E^{u}\). Here, the center subbundle \(E^c\) may contract or expand slightly, but it is dominated by the strong expansion and contraction of the unstable and stable subbundles.NEWLINENEWLINEUnlike the stable and unstable directions, the center badly behaves. So giving a classification of the partially hyperbolic diffeomorphisms, even for special cases, is a difficult task. In this paper, the author applies the techniques of Global Product Structure and the Central Shadowing Lemma to show that every partially hyperbolic diffeomorphism on a 3-dimensional nilmanifold is leaf conjugate to a nilmanifold automorphism. Moreover, if the nilmanifold is not the 3-torus, the center foliation is an invariant circle bundle.