On 3-manifolds that support partially hyperbolic diffeomorphisms
Abstract
Let M be a closed 3-manifold that supports a partially hyperbolic diffeomorphism f. If π1(M) is nilpotent, the induced action of f* on H_1(M, \mathbb{R}) is partially hyperbolic. If π1(M) is almost nilpotent or if π1(M) has subexponential growth, M is finitely covered by a circle bundle over the torus. If π1(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 π1(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.- Publication:
-
Nonlinearity
- Pub Date:
- March 2010
- DOI:
- 10.1088/0951-7715/23/3/009
- arXiv:
- arXiv:1001.1029
- Bibcode:
- 2010Nonli..23..589P
- Keywords:
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- Mathematics - Dynamical Systems;
- 34C40
- E-Print:
- 21 pages