A pasting lemma and some applications for conservative systems
Alexander Arbieto ,
Carlos Matheus
Keywords:
Pasting lemma, robust transitivity, dominated splitting
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.
MSC 2000:
37D30 Partially hyperbolic systems and dominated splittings
