On the rationality of the Nielsen zeta function for maps on solvmanifolds

IF 16.4 1区化学 Q1 CHEMISTRY, MULTIDISCIPLINARY

Accounts of Chemical Research Pub Date : 2024-06-19 DOI:10.1007/s11784-024-01116-9

Karel Dekimpe, Iris Van den Bussche

引用次数: 0

Abstract

In Dekimpe and Dugardein (J Fixed Point Theory Appl 17:355–370, 2015), Fel’shtyn and Lee (Topol Appl 181:62–103, 2015), the Nielsen zeta function \(N_f(z)\) has been shown to be rational if f is a self-map of an infra-solvmanifold of type (R). It is, however, still unknown whether \(N_f(z)\) is rational for self-maps on solvmanifolds. In this paper, we prove that \(N_f(z)\) is rational if f is a self-map of a (compact) solvmanifold of dimension \(\le 5\). In any dimension, we show additionally that \(N_f(z)\) is rational if f is a self-map of an \(\mathcal{N}\mathcal{R}\)-solvmanifold or a solvmanifold with fundamental group of the form \(\mathbb {Z}^n\rtimes \mathbb {Z}\).

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论求解曼菲尔德上映射的尼尔森泽塔函数的合理性

在 Dekimpe 和 Dugardein (J Fixed Point Theory Appl 17:355-370, 2015)、Fel'shtyn 和 Lee (Topol Appl 181:62-103, 2015)中，如果 f 是类型 (R) 的下溶域曼ifold 的自映射，尼尔森zeta函数\(N_f(z)\)被证明是合理的。然而，对于溶点上的自映射，(N_f(z)\) 是否有理仍是未知数。在本文中，我们证明了如果f是维数（\le 5\）的（紧凑）solvmanifold的自映射，那么\(N_f(z)\)就是合理的。在任意维度上，我们还证明了如果 f 是一个 \(\mathcal{N}\mathcal{R}\)- solvmanifold 的自映射，或者是一个基本群形式为 \(\mathbb {Z}^n\rtimes \mathbb {Z}/)的 solvmanifold，那么 \(N_f(z)\ 就是有理的。）

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Accounts of Chemical Research 化学-化学综合

CiteScore

31.40

自引率

1.10%

发文量

312

审稿时长

2 months

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