A pasting lemma and some applications for conservative systems

Abstract

We prove that in a compact manifold of dimension $n\geq 2$, a $C^{1+\alpha}$ volume-preserving diffeomorphisms that are robustly transitive in the $C^1$-topology have a dominated splitting. Also we prove that for 3-dimensional compact manifolds, an isolated robustly transitive invariant set for a divergence-free vector field can not have a singularity. In particular, we prove that robustly transitive divergence-free vector fields in 3-dimensional manifolds are Anosov. For this, we prove some ``pasting'' lemma, which allows to make perturbations in conservative systems.