Nielsen type numbers for periodic points on pairs of spaces

Two homotopy invariant Nielsen type numbers exist for periodic points of a self-map ƒ: X → X of a compact connected ANR X, namely NPn(ƒ), the Nielsen type number of period n, and NΦn(ƒ), the Nielsen type number of the nth iterate. The first is a lower bound for the number of periodic points of least period n, and the second a lower bound for the number of periodic points of all periods m ⩽ n. Both these Nielsen type numbers are extended here to the case of maps ƒ: (X, A) → (X, A) of pairs of compact ANRʹs. Calculations and many examples are given.