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

Abstract

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 $p$.

17 pages