Wecken’s theorem for periodic points in dimension at least 3. (English) Zbl 1094.55005
Boju Jiang [Contemp. Math. 14 (1983; Zbl 0512.55003)] introduced a homotopy invariant \(NF_n(f)\) which is a lower bound for the cardinality of periodic points of period \(n\) for a self-map \(f\) of a compact polyhedron. The present author in [Topology 42, 1101–1124 (2003; Zbl 1026.55001) and “Wecken’s theorem for fixed and periodic points”, in Handbook of Topological fixed point theory, 555–616 (2005; Zbl 1079.55007)] using rather tricky geometric arguments showed that any selfmap \(f:M\to M\) of a compact PL-manifold of dimension at least 3 is homotopic to a map \(g\) such that \(g^n\) has precisely \(NF_n(f)\) fixed points. Here, he gives a much simpler proof of this result adapting a method due to Boju Jiang [Lect. Notes Math. 886, 163–170 (1981; Zbl 0482.57014)] to the case of periodic points.
Reviewer: Christian Fenske (Gießen)
MSC:
| 55M20 | Fixed points and coincidences in algebraic topology |
| 37C25 | Fixed points and periodic points of dynamical systems; fixed-point index theory; local dynamics |
References:
| [1] | Babenko, I. K.; Bogatyi, S. A., Behaviour of the index of periodic points under iterations of a mapping, Izv. Akad. Nauk SSSR Ser. Mat.. Izv. Akad. Nauk SSSR Ser. Mat., Math. USSR-Izv., 38, 1, 1-26 (1992), (in Russian); English translation · Zbl 0742.58027 |
| [2] | Dold, A., Fixed point indices of iterated maps, Invent. Math., 74, 419-435 (1983) · Zbl 0583.55001 |
| [3] | Jezierski, J., Cancelling periodic points, Math. Ann., 321, 107-130 (2001) · Zbl 0994.55003 |
| [4] | Jezierski, J., Wecken theorem for periodic points, Topology, 42, 5, 1101-1124 (2003) · Zbl 1026.55001 |
| [5] | Jezierski, J., Weak Wecken’s Theorem for periodic points in dimension 3, Fund. Math., 160, 223-239 (2003) · Zbl 1052.55003 |
| [6] | Jezierski, J., Wecken theorem for fixed and periodic points, (Handbook of Topological Fixed Point Theory (2005), Kluwer Academic: Kluwer Academic Dordrecht) · Zbl 1079.55007 |
| [7] | Jiang, B. J., Lectures on the Nielsen Fixed Point Theory, Contemp. Math., vol. 14 (1983), American Mathematical Society: American Mathematical Society Providence, RI · Zbl 0512.55003 |
| [8] | Jiang, B. J., Fixed point classes from a differential viewpoint, (Lecture Notes in Math., vol. 886 (1981), Springer: Springer Berlin), 163-170 · Zbl 0482.57014 |
| [9] | Jiang, B. J., Fixed points and braids, Invent. Math., 75, 69-74 (1984) · Zbl 0565.55005 |
| [10] | Nielsen, J., Über die Minimalzahl der Fixpunkte bei Abbildungstypen der Ringflachen, Math. Ann., 82, 83-90 (1921) · JFM 47.0527.03 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
