Sub-shadowing and specification through pointwise dynamics

In this paper, the notion of d̄-shadowable points for continuous maps on compact metric spaces is introduced and it is proved that an equicontinuous map (not necessarily surjective) cannot have the d̄-shadowing property. Consequently, it is shown that a minimal continuous map with the d̄-shadowing property is chaotic in the sense of Auslander-Yorke. Then, it is proved that the existence of a pseudo-orbital specification point, the existence of an ergodic shadowable point, the pseudo-orbital specification property and the ergodic shadowing property are equivalent. Consequently, it is shown that existence of a pseudo-orbital specification point implies the specification property. Next, it is proved that the existence of a point which is both shadowable and specification point implies the pseudo-orbital specification property, the ergodic shadowing property, the shadowing property and the d̄-shadowing property. Consequently, it is shown that the existence of a shadowable point in a topologically mixing system implies the shadowing property as well as the specification property. Finally, it is proved that a continuous map with the specification property is mean sensitive.