Fixed points on pairs of nilmanifolds

A theorem of D. Anosov states that, for any selfmap f: M → M of a compact nilmanifold M, N(f) = ¦L(f)¦ where N(f) and L(f) denote the Nielsen and the Lefschetz numbers of f, respectively. We generalize this result for relative Nielsen type numbers to selfmaps of pairs of nilmanifolds. As an application, we estimate the minimal number of periodic points of prime power period.