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A pasting lemma and some applications for conservative systems

Alexander Arbieto·Carlos Matheus
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Abstract

We prove that in a compact manifold of dimension $n\geq 2$, a $C^{1+α}$ 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.

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Alexander Arbieto, Carlos Matheus. 2006-01-18. A pasting lemma and some applications for conservative systems. https://arxiv.org/abs/math/0601433

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