Preprint A244/2003
A pasting lemma and some applications for conservative systems

Carlos Matheus | Arbieto, Alexander

**Keywords: **
Pasting lemma | robust transitivity | Dominated splitting

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.