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Let $M$ be a manifold with a volume form $\omega$ and $f : M \to M$ be adiffeomorphism 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 mainthemes in the paper: (1) the chain recurrence; (2) relations among recurrencepoints, homoclinic points, shadowability and hyperbolicity. For (1) (withoutassuming $M$ 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 isautomatic. For (2) (assuming the compactness of $M$), we prove some variousimplications among notions, such as: (i) the $\mathcal{C}^1$-stableshadowability equals to the hyperbolicity of $M$; (ii) if a point $p\in M$ hasa recurrence point in the unstable manifold $W^u (p, f)$ and there is nohomoclinic point of $p,$ then $f$ is nonshadowable; (iii) if $f$ has theshadowing property and $p$ has a recurrence point in $W^u (p, f),$ then therecurrent point is in the limit set of homoclinic points of $p$.

On Homoclinic points, Recurrences and Chain recurrences of volume-preserving diffeomorphisms without genericity