Chain Transitive Sets and Shadowing

Abstract

In this chapter, we study relations between the shadowing property of diffeomorphisms on their chain transitive sets and the hyperbolicity of such sets. We prove the following two main results:

• Let Λ be a closed invariant set of f ∈ Diff¹(M). Then f | _Λ is chain transitive and C ¹-stably shadowing in a neighborhood of Λ if and only if Λ is a hyperbolic basic set (Theorem 4.2.1);

• there is a residual set $\mathcal{R} \subset \text{Diff}^{1}(M)$ such that if $f \in \mathcal{ R}$ and Λ is a locally maximal chain transitive set of f, then Λ is hyperbolic if and only if f | _Λ is shadowing (Theorem 4.3.1).