A note on the set of periods for Klein bottle maps
type: Item
instance of: scholarly article
title: A note on the set of periods for Klein bottle maps
zbMATH Open document ID: 0832.55003
author: Jaume Llibre
published in: Pacific Journal of Mathematics
publication date: 1996-03-07T00:00:00Z
review text: By using the Nielsen fixed point theory we characterize the set of periods attained by continuous self-maps on the Klein bottle belonging to a given homotopy class. The main result can be stated as follows (here \(2\mathbb{N}\) denotes the set of all even natural numbers): Theorem. Let \(f\) be a Klein bottle map and let \(u\) and \(v\) be its associated integers. (1) Suppose \(|u|>1\). Then \[ \begin{alignedat}{2} \text{Per} (f)&= \mathbb{N} && \text{if }|v|\neq 1, \qquad \qquad \text{and}\\ \text{Per} (f) &\supset \mathbb{N} \smallsetminus 2\mathbb{N} \qquad &&\text{if }v= - 1.\end{alignedat} \] Furthermore, the last result cannot be improved without additional hypothesis on \(f\). (2) Suppose \(|u|\leq 1\). Then \[ \begin{alignedat}{2} \text{Per} (f)&= \mathbb{N} &&\text{ if } v\not\in \{-2, -1, 0,1\}, \\ \text{Per} (f) &\supset N\smallsetminus \{2\} \qquad &&\text{ if } v=-2, \\ \text{Per} (f) &\supset \{1\} &&\text{ if } v\in \{-1, 0\}.\end{alignedat} \] Furthermore, these last two results cannot be improved without additional hypotheses on \(f\). (3) If \(v=1\) then the set \(\text{Per} (f)\) can be empty. The proof is based on the results of \textit{B. Halpern} [Periodic points on the Klein bottle (Preprint)]\ on the Nielsen numbers of Klein bottle maps. Unfortunately this nice paper of Halpern remains unpublished. As we shall see the Nielsen numbers of the Klein bottle map \(f\) only depend on its associated integers. Then, since the Nielsen numbers are homotopy invariants, the characterization of the set of periods given in the theorem works for all Klein bottle maps homotopic to \(f\).
Mathematics Subject Classification ID: 55M20
Mathematics Subject Classification ID: 57N05
zbMATH DE Number: 508853
zbMATH Keywords: Nielsen fixed point theory
zbMATH Keywords: periods attained by continuous self-maps on the Klein bottle
MaRDI profile type: Publication
full work available at URL: pjm.1993.157.87
OpenAlex ID: W1965515015
DOI: 10.2140/PJM.1993.157.87
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description: scientific article; zbMATH DE number 508853