A Nielsen type number of fibre preserving maps
type: Item
instance of: scholarly article
zbMATH Open document ID: 1273.55002
DOI: 10.4134/CKMS.2013.28.2.361
author: Seoung Ho Lee
published in: Communications of the Korean Mathematical Society
publication date: 2013-08-13T00:00:00Z
full work available at URL: http://www.mathnet.or.kr/mathnet/kms_content.php
Mathematics Subject Classification ID: 55M20
Mathematics Subject Classification ID: 54H25
zbMATH DE Number: 6198212
zbMATH Keywords: essential \(n\)-orbit
zbMATH Keywords: Nielsen type number
zbMATH Keywords: fibre space
MaRDI profile type: Publication
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title: A Nielsen type number of fibre preserving maps
review text: Nielsen fixed point theory deals with the estimation of the numbers of fixed points, and has been used to deal with those of periodic points. The author of the paper under review considers the periodic points of fibre-preserving maps on fibre spaces.NEWLINENEWLINEGiven a fibre-preserving map \(f\) on a fibre space \(p: E\to B\), one has a reduced map \(\bar f: B \to B\) on the base space \(B\). In this situation, the author defines an invariant for such a map \(f\) which is a lower bound for the number of periodic \(n\)-orbits of \(f\), under the assumption that \(f\) is essentially \(n\)-orbit fibre uniform. Roughly speaking, this condition means that the essential \(n\)-orbit on each fibre over periodic points of \(\bar f\) are similar. It is also shown that this invariant is equal to a relative Nielsen type number for periodic \(n\)-orbits which was introduced in [\textit{P. R. Heath, H. Schirmer} and \textit{C. You}, Topology Appl. 63, No. 2, 117--138 (1995; Zbl 0827.55002)]. Finally, the author proves that under some more assumptions, his new invariant is the same as the usual Nielsen type number for periodic points.
reviewed by: Xuezhi Zhao
description: scientific article; zbMATH DE number 6198212