### Wecken property for periodic points on the Klein bottle

DOI: http://dx.doi.org/10.12775/TMNA.2009.005

#### 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 $\# \text{\rm Fix}(g^m)$

over all $g\sim f$. This question has a positive answer if

$\text{\rm dim} M\geq 3$ 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

$\text{\rm HPer} (f)$ of homotopy minimal periods on the Klein bottle.

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 $\# \text{\rm Fix}(g^m)$

over all $g\sim f$. This question has a positive answer if

$\text{\rm dim} M\geq 3$ 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

$\text{\rm HPer} (f)$ of homotopy minimal periods on the Klein bottle.

#### Keywords

Periodic point; homotopy minimal period; Nielsen number; Nielsen-Jiang periodic number; Klein bottle

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