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A pasting lemma and some applications for conservative systems

Published online by Cambridge University Press:  01 October 2007

ALEXANDER ARBIETO  and
CARLOS MATHEUS
ALEXANDER ARBIETO
Affiliation:
IMPA, Estrada D. Castorina 110, Jardim Botânico, 22460-320 Rio de Janeiro, Brazil (email: alexande@impa.br, cmateus@impa.br)
CARLOS MATHEUS
Affiliation:
IMPA, Estrada D. Castorina 110, Jardim Botânico, 22460-320 Rio de Janeiro, Brazil (email: alexande@impa.br, cmateus@impa.br)
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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.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 27 , Issue 5 , October 2007 , pp. 1399 - 1417
Copyright
Copyright © Cambridge University Press 2007

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