HYPERBOLICITY OF CHAIN TRANSITIVE SETS WITH LIMIT SHADOWING

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