LAUSR

We study the eventually shadowable points namely points for which every pseudo orbit passing through then can be eventually shadowed (Good and Meddaugh in Ergod Theory Dyn Syst 38(1):143–154, 2018). We will prove the following results: the set of eventually shadowable points of a surjective continuous map of a compact metric space is invariant (possibly empty or noncompact) and the map has the eventual shadowing property if and only if every point is eventually shadowable. The chain recurrent and nonwandering sets coincide when every chain recurrent point is eventually shadowable. A surjective continuous map of a compact metric space has the eventual shadowing property if and only if the set of eventually shadowable points has a full measure with respect to every ergodic invariant probability measure. If there is an eventually shadowable point for which the associated Li–Yorke set equals the whole space, then the map has the eventual shadowing property. Proximal or transitive maps with eventually shadowable points have the eventual shadowing property. The eventually shadowable and shadowable points coincide for surjective equicontinuous maps on compact metric spaces. In particular, a surjective equicontinuous map of a compact metric space has the eventual shadowing property if and only if it has the shadowing property.