Dynamical Property of Family on Nd

Blohk and Fieldsteel introduced firstly the concept of set forcing recurrence in 2002. A subset S of the set of positive integers N is called a set forcing recurrence, if for any dynamical system (X, f) and any compact set K  X, whenever there exists x 2 X such that {fnx|n 2 S}  K, then K must contain some recurrence points of f. In this paper, the definition of set of recurrence times of Nd (d 2 N) is introduced, and several characters of the families of sets of recurrence times are obtained. Furthermore, an application of the sets of recurrence times is provided.