The crucial homotopy invariants in Nielsen periodic point theory are numbers: <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11784_2024_1155_Article_IEq3.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="58" /> </InlineMediaObject> <EquationSource Format="TEX">$NP_n(f)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>N</mi> <msub> <mi>P</mi> <mi>n</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, which is a lower bound of the number of periodic points of length n, and <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11784_2024_1155_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="58" /> </InlineMediaObject> <EquationSource Format="TEX">$NF_n(f)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>N</mi> <msub> <mi>F</mi> <mi>n</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> a lower bound of the number of periodic points of length dividing n. Here, <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11784_2024_1155_Article_IEq5.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="86" /> </InlineMediaObject> <EquationSource Format="TEX">$f:X\rightarrow X$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo>:</mo> <mi>X</mi> <mo stretchy="false">→</mo> <mi>X</mi> </mrow> </math></EquationSource> </InlineEquation> is a self-map of a compact polyhedron. We derive formulae of the invariants for self-maps of polyhedra with fundamental group <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11784_2024_1155_Article_IEq6.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="80" /> </InlineMediaObject> <EquationSource Format="TEX">$\pi _1 M=\mathbb {Z}_{p^s}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>π</mi> <mn>1</mn> </msub> <mi>M</mi> <mo>=</mo> <msub> <mi mathvariant="double-struck">Z</mi> <msup> <mi>p</mi> <mi>s</mi> </msup> </msub> </mrow> </math></EquationSource> </InlineEquation> whose all irreducible classes are essential.

Periodic points of self-maps of a space with $\pi _1X=\mathbb {Z}_{p^s}$

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发表日期：

2025年1月9日

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Jerzy Jezierski,
Xuezhi Zhao

摘要

The crucial homotopy invariants in Nielsen periodic point theory are numbers: $NP_n(f)$ $N P_{n} (f)$ , which is a lower bound of the number of periodic points of length n, and $NF_n(f)$ $N F_{n} (f)$ a lower bound of the number of periodic points of length dividing n. Here, $f:X\rightarrow X$ $f : X \to X$ is a self-map of a compact polyhedron. We derive formulae of the invariants for self-maps of polyhedra with fundamental group $\pi _1 M=\mathbb {Z}_{p^s}$ $π_{1} M = Z_{p^{s}}$ whose all irreducible classes are essential.

Periodic points of self-maps of a space with \(\pi _1X=\mathbb {Z}_{p^s}\)

摘要