2010/01/07 by Kamlesh Parwani · 15 citations
Mathematics · #Action (physics) #Analytic and geometric function theory #Bundle #Diffeomorphism #Geometric and Algebraic Topology #Hyperbolic manifold #Hyperbolic set #Mathematical Dynamics and Fractals #Nilpotent #Relatively hyperbolic group #Stable manifold #Torus #math.DS #msc:34C40
paper · pdf · doi:10.1088/0951-7715/23/3/009
published in Nonlinearity 23(3), 589-606 (IOP Publishing) · 21 pages
arxiv created 2010/01/07 · openalex publication_date 2010/01/21 · arxiv updated 2015/05/14 · openalex created_date 2016/06/24 · openalex updated_date 2026/08/05
Let M be a closed 3-manifold that supports a partially hyperbolic diffeomorphism f . If π 1 ( M ) is nilpotent, the induced action of f * on is partially hyperbolic. If π 1 ( M ) is almost nilpotent or if π 1 ( M ) has subexponential growth, M is finitely covered by a circle bundle over the torus. If π 1 ( M ) is almost solvable, M is finitely covered by a torus bundle over the circle. Furthermore, there exist infinitely many hyperbolic 3-manifolds that do not support dynamically coherent partially hyperbolic diffeomorphisms; this list includes the Weeks manifold. If f is a strong partially hyperbolic diffeomorphism on a closed 3-manifold M and if π 1 ( M ) is nilpotent, then the lifts of the stable and unstable foliations are quasi-isometric in the universal cover of M . It then follows that f is dynamically coherent. We also provide a sufficient condition for dynamical coherence in any dimension. If f is centre-bunched and if the centre-stable and centre-unstable distributions are Lipschitz, then the partially hyperbolic diffeomorphism f must be dynamically coherent.