Nielsen fixed point theory deals with the estimation of the number of fixed points. Given a map \(f: X\to X\) on a compact polyhedron, the so-called Nielsen number \(N(f)\) gives a lower bound for the number of fixed points of all maps in the homotopy class of \(f\). Using a similar idea, B. Jiang introduced another invariant \(NF_r(f)\) (or say \(N\Phi_r(f)\)), giving a lower bound for the number of fixed points of \(r\)-iterations all maps in the homotopy class of \(f\). In general, \(NF_r(f)\geq N(f^r)\) for any positive integer \(r\).
The paper under review deals with the estimation of the number \(\sharp Fix(f^r)\) of fixed points of \(f^r\) when \(f\) is a smooth map. A new invariant \(NJD_r[f]\) is defined for any positive integer \(r\), with the property \(NJD_r[f]\leq \sharp Fix(f^r)\). It is proved that \(NJD_r[f]\) can be reached if \(f\) is a smooth map on a smooth manifold of dimension at least \(3\), i.e for any \(r\), there will be a smooth map \(g\) homotopic to \(f\) such that \(\sharp Fix(g^r) = NJD_r[f]\). The method is similar to that in the authors’ previous work [Forum Math. 21, No. 3, 491–509 (2009; Zbl 1173.37014)]. A key point is the observation of S.-N. Chow, J. Mallet-Paret and J. A. Yorke [Lect. Notes Math. 1007, 109–131 (1983; Zbl 0549.34045)]: there exist some inter-relations among the fixed point indices of all iterations of an isolated fixed point of a \(C^1\) map. This leads to an interesting phenomenon: smooth maps contain more periodic points than continuous ones in a given homotopy class. Here, more techniques are involved because the underlying manifold is no longer simply-connected hence fixed point classes have to be taken into account. The authors of this paper handle successfully and precisely such a difference. Some concrete examples, maps on \(\mathbb RP^3\), are given, showing that \(NJD_r[f]\) can be greater than \(NF_r(f)\).