Nielsen numbers of iterates and Nielsen type periodic numbers of periodic maps on tori and nilmanifolds

In this paper we compute the Nielsen numbers N(f(m)) and the Nielsen type numbers NPm (f) and N Phi(m) (f) for all m, for periodic maps f on tori and nilmanifolds. For fixed m, there are known formulas for these numbers for arbitrary maps on tori and nilmanifolds. However when seeking to determine these numbers for all m for periodic maps, fascinating patterns and shortcuts are revealed. Our method has two main thrusts. Firstly we study N(f(m)),NPm(f) and N Phi(m) (f) on primitives (maps whose linearizations consist of primitive roots of unity), and then secondly we employ fibre techniques to give an inductive approach to the general case adding one primitive at a time. This approach is made possible by the eigen structure of the linearizations of the maps involved.