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The notion of shadowable points was introduced by Morales in his recent paper (2016,Dynamical Systems). A shadowable point of a continuous map or a homeomorphism is defined to be a point such that the shadowing lemma holds for pseudo orbits passing through the point. In this paper, as a quantitative version of the shadowable points, we study shadowable points with a given shadowing accuracy, and prove a quantitative version of Morales’ theorem in his paper above. In addition, we answer two questions on shadowable points asked in the paper. [ABSTRACT FROM PUBLISHER]

Let f be a diffeomorphism of a closed C ∞ manifold M, and let Λ ⊂ M be a closed f-invariant set. We show that if f |Λ is robustly chain transitive with orbital shadowing property, then Λ is a basic set. [ABSTRACT FROM PUBLISHER]

Let Diff(M) be the space of diffeomorphisms of a closed C ∞ manifold M endowed with the C 1-topology, and denote by ℛ(f) the chain recurrent set of f ∈ Diff(M). In this article, we show that C 1-generically f |ℛ(f) has the s-limit shadowing property if and only if f satisfies both Axiom A and no-cycle condition. [ABSTRACT FROM PUBLISHER]