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

Abstract

In this paper we compute the Nielsen numbers $N(f^m)$ and the Nielsen type numbers $NP_m(f)$ and $N\Phi_m(f)$ {\it 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 $N(f^m)$, $NP_m(f)$ 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.