A Nielsen type number of fibre preserving maps
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.
scientific article; zbMATH DE number 6198212