A Pasting Lemma I: the case of vector fields

We prove a perturbation (pasting) lemma for conservative (and symplectic) systems. This allows us to prove that $C^{\infty}$ volume preserving vector fields are $C^1$-dense in $C^{1}$ volume preserving vector fields (After the conclusion of this work, Ali Tahzibi pointed out to us that this result was proved in 1979 by Carlos Zuppa, although his proof is different from ours.). Moreover, we obtain that $C^1$ robustly transitive conservative flows in three-dimensional manifolds are Anosov and we conclude that there are no geometrical Lorenz-like sets for conservative flows. Also, by-product of the version of our pasting lemma for conservative diffeomorphisms, we show that $C^1$-robustly transitive conservative $C^2$-diffeomorphisms admits a dominated splitting, thus solving a question posed by Bonatti-Diaz-Pujals. In particular, stably ergodic diffeomorphisms admits a dominated splitting.