• Communications of the Korean Mathematical Society
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• v.27 no.2
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• pp.369-375
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• 2012

In this article, we focus on certain dynamic phenomena in volume-preserving manifolds. Let $M$ be a compact manifold with a volume form ${\omega}$ and $f:M{\rightarrow}M$ be a diffeomorphism of class $\mathcal{C}^1$ that preserves ${\omega}$. In this paper, we do not assume $f$ is $\mathcal{C}^1$-generic. We prove that $f$ satisfies the chain transitivity and we also show that, on $M$, the $\mathcal{C}^1$-stable shadowability is equivalent to the hyperbolicity.