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december 2017 Nielsen numbers of iterates and Nielsen type periodic numbers of periodic maps on tori and nilmanifolds
Philip R. Heath
Bull. Belg. Math. Soc. Simon Stevin 24(4): 689-723 (december 2017). DOI: 10.36045/bbms/1515035017

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.

Citation

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Philip R. Heath. "Nielsen numbers of iterates and Nielsen type periodic numbers of periodic maps on tori and nilmanifolds." Bull. Belg. Math. Soc. Simon Stevin 24 (4) 689 - 723, december 2017. https://doi.org/10.36045/bbms/1515035017

Information

Published: december 2017
First available in Project Euclid: 4 January 2018

zbMATH: 06848711
MathSciNet: MR3743272
Digital Object Identifier: 10.36045/bbms/1515035017

Subjects:
Primary: ‎55M20

Rights: Copyright © 2017 The Belgian Mathematical Society

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Vol.24 • No. 4 • december 2017
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