Least number of \(n\)-periodic points of self-maps of \(PSU(2)\times PSU(2)\)
type: Item
instance of: scholarly article
title: Least number of \(n\)-periodic points of self-maps of \(PSU(2)\times PSU(2)\)
zbMATH Open document ID: 1494.55002
author: Jerzy Jezierski
published in: Journal of Fixed Point Theory and Applications
publication date: 2022-02-02T00:00:00Z
review text: Let \(f : M \to M\) be a self-map of a compact manifold and \(n \in \mathbb{N}.\) \textit{B. Jiang} [Lectures on Nielsen fixed point theory. Providence, RI: American Mathematical Society (AMS) (1983; Zbl 0512.55003)] introduced a number, \(NF_n(f),\) called Nielsen-Jiang periodic number of \(f.\) Also, G. Graff and J. Jezierski introduced a number, \(NJD_n(f),\) called Nielsen-Jiang-Dold number of \(f\), [\textit{G. Graff} and \textit{J. Jezierski}, Topology Appl. 158, No. 3, 276--290 (2011; Zbl 1211.55004)]. If \(dim(M) \geq 3\) then these numbers satisfy \(NF_n(f) = \min\{\#Fix(g^{n})\mid g \sim f, \,\, g \,\, continuous \}\) and \(NJD_n(f) = min\{\#Fix(g^{n})\mid g \sim f, \,\, g \,\, smooth \}.\) In general \(NF_n(f) < NJD_n(f).\) The main question is to know when \(NF_n(f) = NJD_n(f)\)? If \(f\) is essentially reducible, has the Jiang property and \(L(f^{k}) = det(I - A^{k})\) for an integer matrix \(A,\) then a necessary condition for the equality \(NF_n(f) = NJD_n(f)\) can be shown. In this paper the author considers \(M = PSU(2),\) the projective unitary group, or \(M = PSU(2) \times PSU(2),\) and shows that this necessary condition is also sufficient. The proof of this fact is made using the description of the graph of orbits of Reidemeister classes \(\mathcal{G} \mathcal{O} \mathcal{R}(f^{n}).\)
reviewed by: Weslem L. Silva
Mathematics Subject Classification ID: 55M20
Mathematics Subject Classification ID: 37C05
Mathematics Subject Classification ID: 37C25
zbMATH DE Number: 7468365
zbMATH Keywords: periodic points
zbMATH Keywords: Nielsen number
zbMATH Keywords: fixed point index
zbMATH Keywords: smooth maps
zbMATH Keywords: Lie group
MaRDI profile type: Publication
full work available at URL: s11784-021-00921-w
OpenAlex ID: W4200038002
Wikidata QID: Q114221814
cites work: THE BEHAVIOR OF THE INDEX OF PERIODIC POINTS UNDER ITERATIONS OF A MAPPING
cites work: Q5619504
cites work: Q3342939
cites work: Fixed point indices of iterated maps
cites work: The Lefschetz numbers of iterated maps
cites work: Über die Topologie der Gruppen-Mannigfaltigkeiten und ihre Verallgemeinerungen
cites work: Minimal number of periodic points for C 1 self-maps of compact simply-connected manifolds
cites work: Minimizing the number of periodic points for smooth maps. Non-simply connected case
cites work: Fixed point indices of iterated smooth maps in arbitrary dimension
cites work: Fibre techniques in Nielsen periodic point theory on nil and solvmanifolds. I
cites work: Model solvmanifolds for Lefschetz and Nielsen theories
cites work: Least number of periodic points of self-maps of Lie groups
cites work: A sufficient condition for the realizability of the least number of periodic points of a smooth map
cites work: When a smooth self-map of a semi-simple Lie group can realize the least number of periodic points
cites work: Q5713425
cites work: The least number, of \(n\)-periodic points of a self-map of a solvmanifold, can be realised by a smooth map
cites work: Lectures on Nielsen fixed point theory
cites work: A remark on the Lefschetz fixed point formula for differentiable maps
cites work: Q4847408
DOI: 10.1007/S11784-021-00921-W
description: scientific article; zbMATH DE number 7468365