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 $f:M{ ightarrow}M$ 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.