on Dynamical Systems and Related Topics c©2008 ICMS in KAIST C1-STABLY WEAK SHADOWING CHAIN COMPONENTS ARE PARTIALLY HYPERBOLIC

Abstract. Let f be a diffeomorphism of a closed C ∞ three-dimensional man-ifold. In this paper, we introduce the notion of C1-stably weak shadowing for a closed f-invariant set, and prove that C1-generically, for an aperiodic chain component Cf of f isolated in the chain recurrent set, if f|Cf is C 1-stably weak shadowing, then there are a C1-neighborhood U(f) of f and an open and dense subset V of U(f) such that for any g ∈ V, there is a chain component (of g nearby Cf) which is partially hyperbolic. 1. Introduction. The weak shadowing property of dynamical systems was introduced in [4], and it was shown that this property is C0-generic in the space of diffeomorphisms of a closed C ∞ manifold. Of course, every homeomorphism having the shadowing property has the weak shadowing property, but its converse is not true. Indeed, a