The implication of eventually shadowing property on a ℤd-action to the equivalence of chain recurrent set and omega limit set

According to observation from the past research has triggered a compulsive idea to define a concept of eventually shadowing property for a ℤd-action. By letting the ℤd-action as a map T : ℤd × X → X on a compact metric space (X,ρ), this study observes the two type of sets with certain traits which are known as the k-type omega limit set, Ωk(T) and the k-type chain recurrent set, CRk (T). Some particular connections need to be elaborated before approaching the main objective of this study which is to determine that Ωk(T) = CRk (T) if the map T has eventually shadowing property on the set CRk (T) and k-type weak extending property for some k ∈ {1, 2, …, 2d}. Lastly, this study shows the comparison between the discovery of the above result for the ℤd-action with the findings of the similar implications which were already done before in the case of homeomorphism.