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

TL;DR

This paper investigates the dynamics of volume-preserving diffeomorphisms without assuming genericity, focusing on chain recurrence, homoclinic points, shadowability, and hyperbolicity, revealing new relations among these concepts.

Contribution

It establishes new results on chain recurrence and the interplay between recurrence points, homoclinic points, shadowability, and hyperbolicity for volume-preserving diffeomorphisms without genericity assumptions.

Findings

If f is Lagrange stable, then M is a chain recurrent set.

Stable shadowability is equivalent to hyperbolicity of M.

Presence of recurrence points without homoclinic points implies nonshadowability.

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 $p$.