A pasting lemma and some applications for conservative systems

Abstract

We prove that in a compact manifold of dimension n≥2, C1+α volume-preserving diffeomorphisms that are robustly transitive in the C1-topology have a dominated splitting. Also we prove that for three-dimensional compact manifolds, an isolated robustly transitive invariant set for a divergence-free vector field cannot have a singularity. In particular, we prove that robustly transitive divergence-free vector fields in three-dimensional manifolds are Anosov. For this, we prove a ‘pasting’ lemma, which allows us to make perturbations in conservative systems.

Asymptotich-expansiveness rate ofC∞maps

“…The map f a is chosen to be C ∞ flat at 0 so that F a has a homoclinic tangency of infinite order at (0, 0). Moreover since (x, y) → (x, y 1 + f a (x 1 ), y 2 , · · · , y mu ) is volume preserving we may choose θ a be also volume preserving by the pasting Lemma of Arbieto and Matheus (see Lemma 3.9 of [4]). Also θ a is C ∞ close to the identity when f a is C ∞ close to zero.…”

On the regularization of conservative maps

“…Let us point out that the corresponding regularization theorem for conservative flows was obtained much earlier by Zuppa [29] in 1979. In fact, in a more recent approach of Arbieto-Matheus [1], it is shown that a result of Dacorogna-Moser [13] allows one to reduce to a local situation where the regularization of vector fields which are divergence free can be treated by convolutions. However, attempts to reduce the case of maps to the case of flows through a suspension construction have not been successful.…”

C1-STABLY SHADOWABLE CONSERVATIVE DIFFEOMORPHISMS ARE ANOSOV

“…[8]); (2) the result in [14] are replaced by the smoothness result in [3]; (3) the perturbation lemma in [6], are interchanged by its correspondent in the volume-preserving case proved in [5, Proposition 7.4]; (4) and, finally, we should use [2, Theorem 3.6] instead of [2,Theorem 3.9].…”