Abstract
Abstract. In this paper we show that any chain transitive set of a dif-feomorphism on a compact C ∞ -manifold which is C 1 -stably limit shad-owable is hyperbolic. Moreover, it is proved that a locally maximal chaintransitive set of a C 1 -generic diffeomorphism is hyperbolic if and only ifit is limit shadowable. Transitivesets, homoclinicclassesand chaincomponentsofadiffeomorphismare natural candidates to replace the hyperbolic basic sets in nonhyperbolictheory of differentiable dynamical systems, and many recent papers exploredtheir “hyperbolic-like” properties (for more details, see [2, 6, 8, 9, 14, 15]).In this paper we study the chain transitive sets which are limit shadowable.Let us be more precise. Let M be a compact C ∞ -manifold, and let Diff(M) bethe space of diffeomorphisms of M endowed with the C 1 -topology. Denote byd the distance on M induced from a Riemannian metric on the tangent bundleTM. For δ > 0, a sequence {x n } n∈Z in Λ is called a δ-limit chain iflim |n|→∞ d(f(x
Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
