  Let $M$ be a manifold with a volume form $\omega$ and $f : M \to M$ be a
diffeomorphism of class $\mathcal{C}^1$ that preserves $\omega$. In this paper,
we do \textit{not} assume $f$ is $\mathcal{C}^1$-generic. We have two main
themes in the paper: (1) the chain recurrence; (2) relations among recurrence
points, homoclinic points, shadowability and hyperbolicity. For (1) (without
assuming $M$ is compact), we have the theorem: if $f$ is Lagrange stable, then
$M$ is a chain recurrent set. If $M$ is compact, then the Lagrange-stability is
automatic. For (2) (assuming the compactness of $M$), we prove some various
implications among notions, such as: (i) the $\mathcal{C}^1$-stable
shadowability equals to the hyperbolicity of $M$; (ii) if a point $p\in M$ has
a recurrence point in the unstable manifold $W^u (p, f)$ and there is no
homoclinic point of $p,$ then $f$ is nonshadowable; (iii) if $f$ has the
shadowing property and $p$ has a recurrence point in $W^u (p, f),$ then the
recurrent point is in the limit set of homoclinic points of $p$.
