Topological Methods in Nonlinear Analysis

Wecken property for periodic points on the Klein bottle

Jerzy Jezierski, Edward Keppelmann, and Wacław Marzantowicz

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Abstract

Suppose $f\colon M\to M$ on a compact manifold. Let $m$ be a natural number. One of the most important questions in the topological theory of periodic points is whether the Nielsen-Jiang periodic number $NF_m(f)$ is a sharp lower bound on $\# {\rm Fix}(g^m)$ over all $g\sim f$. This question has a positive answer if ${\rm dim}\, M\geq 3$ but in general a negative answer for self maps of compact surfaces. However, we show the answer to be positive when $M=\mathbb K$ is the Klein bottle. As a consequence, we reconfirm a result of Llibre and compute the set ${\rm HPer} (f)$ of homotopy minimal periods on the Klein bottle.

Article information

Source
Topol. Methods Nonlinear Anal., Volume 33, Number 1 (2009), 51-64.

Dates
First available in Project Euclid: 27 April 2016

Permanent link to this document
https://projecteuclid.org/euclid.tmna/1461782239

Mathematical Reviews number (MathSciNet)
MR2512954

Zentralblatt MATH identifier
1179.55003

Citation

Jezierski, Jerzy; Keppelmann, Edward; Marzantowicz, Wacław. Wecken property for periodic points on the Klein bottle. Topol. Methods Nonlinear Anal. 33 (2009), no. 1, 51--64. https://projecteuclid.org/euclid.tmna/1461782239


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