Chain Recurrences on Conservative Dynamics

Abstract

Abstract. Let M be a manifold with a volume form ! and f : M ! M be a difieo-morphism of class C 1 that preserves ! . We prove that if M is almost bounded for thedifieomorphism f , then M is chain recurrent. Moreover, we get that Lagrange stablevolume-preserving manifolds are also chain recurrent. 1. IntroductionOur purpose of this paper is to study the chain recurrence set of volume-preserving difieomorphisms on non-compact manifolds. We follow Conley’s defl-nitions of attractors and chain recurrences [4], and Hurley’s generalized deflnitions[5],[6].From Poincar¶e recurrence theorem, it is well-known that for any volume-preserving difieomorphism on the compact manifolds M , every point of M is chainrecurrent. However, unfortunately, the parallel statement for the chain recurrencedoes not hold for the non-compact manifolds. Thus, in the non-compact case, wemay impose the canonical conditions as almost boundedness and Lagrange stability.Our main theorem is as follows. * Corresponding Author.