Tori with hyperbolic dynamics in 3-manifolds

Abstract

Let M be a closed orientable irreducible 3-dimensional manifold. An embedded 2-torus T is an Anosov torus if there exists a diffeomorphism f over M for which T is f-invariant and f#T: π1 (T) → π1 (T) is hyperbolic. We prove that only few irreducible 3-manifolds admit Anosov tori: (1) the 3-torus T3; (2) the mapping torus of-Id; and (3) the mapping tori of hyperbolic automorphisms of T2. This has consequences for instance in the context of partially hyperbolic dynamics of 3-manifolds: if there is an invariant foliation. Tcu tangent to the center-unstable bundle Ec⊕Eu, then Tcu has no compact leaves [21]. This has led to the first example of a non-dynamically coherent partially hyperbolic diffeomorphism with one-dimensional center bundle [21]. © 2011 AIMSciences.