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

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dc.creatorChoy, Jaeyoo
dc.creatorChu, Hahng-Yun
dc.creatorKim, Min Kyu
dc.date2009-04-07
dc.date.accessioned2026-07-07T13:01:18Z
dc.date.available2026-07-07T13:01:18Z
dc.descriptionLet $M$ be a manifold with a volume form $ω$ and $f : M \to M$ be a diffeomorphism of class $\mathcal{C}^1$ that preserves $ω$. 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 $p$.
dc.description17 pages
dc.identifierhttps://arxiv.org/abs/0904.1142
dc.identifierhttp://arxiv.org/abs/0904.1142
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/226107
dc.subjectDynamical Systems
dc.subjectSymplectic Geometry
dc.subject37C50 (Primary) 37C29; 37D05 (Secondary)
dc.titleOn Homoclinic points, Recurrences and Chain recurrences of volume-preserving diffeomorphisms without genericity
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