Robustly shadowable chain transitive sets and hyperbolicity

We say that a compact invariant set Λ of a C 1 -vector field X on a compact boundaryless Riemannian manifold M is robustly shadowable if it is locally maximal with respect to a neighbourhood U of Λ, and there exists a C 1 -neighbourhood of X such that for any , the continuation Λ Y of Λ for Y and U is shadowable for Y t . In this paper, we prove that any chain transitive set of a C 1 -vector field on M is hyperbolic if and only if it is robustly shadowable.