Surplus Nielsen type numbers for periodic points on the complement

Let f:(X,A)→(X,A) be a self-map of a pair of compact polyhedra with X connected. In 1990 Xuezhi Zhao studied Nielsen fixed point theory on the complement. He defined a surplus Nielsen number SN(f;X−A) which is a good lower bound for the number of fixed points on X−A for all maps in the homotopy class of f. In this paper we generalize these ideas from fixed point theory to periodic point theory, and define two Nielsen type numbers SNPn(f;X−A) and SNΦn(f;X−A) for periodic points on the complement. Thus we answer an open question posed by Heath and Zhao.