Communications of the
Korean Mathematical Society CKMS

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.

Keywords: hyperbolicity, $\mathcal{C}^1$-stable shadowable, chain recurrence, chain transitive

MSC numbers: Primary 37C50; Secondary 37C70, 54H20