A pasting lemma and some applications for conservative systems

We prove that in a compact manifold of dimension n ≥ 2, C ^1+α volume-preserving diffeomorphisms that are robustly transitive in the C¹-topology have a dominated splitting. Also we prove that for three-dimensional compact manifolds, an isolated robustly transitive invariant set for a divergence-free vector field cannot have a singularity. In particular, we prove that robustly transitive divergence-free vector fields in three-dimensional manifolds are Anosov. For this, we prove a 'pasting' lemma, which allows us to make perturbations in conservative systems.