Periodic points on the complement

Let f :(X,A)→(X,A) be a self-map of a pair of compact ANRs, with X connected. In 1989 the second author studied fixed point theory on the complement. He defined a number N(f;X−A) which is a lower bound for the number of fixed points on X−A of maps g that are homotopic to f as a map of pairs. In this paper we generalize these ideas from fixed point theory to periodic point theory, and define two Nielsen type numbers for periodic points on the complement. We give a number of examples and follow a particular one through to show, that for this type of example one of the Nielsen type periodic numbers is given by a formula, and the other is algorithmic. We also highlight a computational simplification of the modified fundamental group approach (which we use here) over the covering space approach.