The least number of 2-periodic points of a smooth self-map of \({S}^{2}\) of degree 2 equals 2
type: Item
instance of: scholarly article
title: The least number of 2-periodic points of a smooth self-map of \({S}^{2}\) of degree 2 equals 2
zbMATH Open document ID: 1486.55004
author: Jerzy Jezierski
published in: Journal of Fixed Point Theory and Applications
publication date: 2019-01-30T00:00:00Z
review text: The purpose of the paper is to show what the title say: The least number of 2-periodic points of a smooth self-map of \(S^2\) of degree 2 equals 2. More precisely: Theorem 3.2: There exists a smooth map \(f: S^2 \to S^2\) of degree 2 which has only two \(2\)-periodic points. Besides proving the result, the author provides a quite detailed and well organised review of the study of the problem of determining the minimal number of periodic points of a given period \(n\). This minimal problem is studied in the context of continuous maps as well as in the context of smooth maps. This is a major problem for surfaces. Some results for self maps of the \(2\)-sphere are known and this new result adds to this case. Since we are in dimension \(2\) the usual Whitney trick of canceling intersection points does not hold in low dimensions. So it is worth reproducing the following comments basically from the author how to obtain his result: ``We get a smooth map directly. We start with a map which is smooth near the poles and we use continuous deformations which are constant near the poles. Finally we deform the obtained map, with only two 2-periodic points, to a smooth map by a homotopy constant near the poles.''
reviewed by: Daciberg Lima Gonçalves
Mathematics Subject Classification ID: 55M20
Mathematics Subject Classification ID: 54H25
Mathematics Subject Classification ID: 57R99
zbMATH DE Number: 7008525
zbMATH Keywords: fixed point
zbMATH Keywords: periodic point
zbMATH Keywords: Nielsen fixed point theory
zbMATH Keywords: Dold congruences
zbMATH Keywords: least number of periodic points
zbMATH Keywords: smooth maps
zbMATH Keywords: the sphere \(S^2\)
zbMATH Keywords: degrees
MaRDI profile type: Publication
full work available at URL: s11784-018-0651-1
OpenAlex ID: W2906659763
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DOI: 10.1007/S11784-018-0651-1
description: scientific article; zbMATH DE number 7008525