Bulletin of the
Korean Mathematical Society
BKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

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Bull. Korean Math. Soc. 2014; 51(1): 67-76

Printed January 1, 2014

https://doi.org/10.4134/BKMS.2014.51.1.67

Copyright © The Korean Mathematical Society.

Shadowing, expansiveness and stability of divergence-free vector fields

C\'elia Ferreira

Universidade do Porto

Abstract

Let $X$ be a divergence-free vector field defined on a closed, connected Riemannian manifold. In this paper, we show the equivalence between the following conditions:
$\bullet$ $X$ is a divergence-free vector field satisfying the shadowing property.
$\bullet$ $X$ is a divergence-free vector field satisfying the Lipschitz shadowing property.
$\bullet$ $X$ is an expansive divergence-free vector field.
$\bullet$ $X$ has no singularities and is Anosov.

Keywords: shadowing, Lipschitz shadowing, expansiveness, Anosov vector fields

MSC numbers: 37C50, 37D20, 37C27, 37C10.