On a structure of the fixed point set of homogeneous maps
type: Item
instance of: scholarly article
title: On a structure of the fixed point set of homogeneous maps
zbMATH Open document ID: 1279.47078
author: Yakov Krasnov
author: Alexander Kononovich
author: G. Osharovich
published in: Discrete and Continuous Dynamical Systems. Series S
publication date: 2013-03-14T00:00:00Z
review text: For an integer \(m\geq 0\), a map \(f:\mathbb R^n\to \mathbb R^n\) is a polynomial \(m\)-homogeneous map if \(f(\mu x)=\mu^mf(x)\) for all \(x\in \mathbb R^n\), \(\mu\in \mathbb R\). The null set of \(f\) is denoted by \(\text{Nul}(f)\) and the set of all non-zero fixed points of \(f\) by \(\text{Fix}(f)\). Let \(V\) be an \(n\)-dimensional real vector space. Let \(\text{Sym}^m(V)\) denote the \(m\)-th symmetric tensor power of \(V\). The paper is concerned with four problems: (1) to find algebraic conditions on sets \(P\) and \(N\) in \(\mathbb C^n\), \(P\cap N=\emptyset\), \(0\in N\), such that there exists a polynomial \(m\)-homogeneous map \(f\) with \(P\subset \text{Fix}(f)\) and \(N\subset \text{Nul}(f)\); (2) in the case that such an \(f\) exists, to determine when it is unique; (3) to extend the sets \(P\) and \(N\) to the spectrum \(S_f:=\text{Fix}(f)\cup\text{Nul}(f)\) of \(f\) without giving and explicit construction of \(f\); (4) to study the topological and dynamical properties of a polynomial \(m\)-homogeneous map \(f\) determined by its spectrum \(S_f\).
Mathematics Subject Classification ID: 47H60
Mathematics Subject Classification ID: 34A34
Mathematics Subject Classification ID: 15A69
Mathematics Subject Classification ID: 15A18
Mathematics Subject Classification ID: 34C14
Mathematics Subject Classification ID: 34C41
zbMATH DE Number: 6144761
zbMATH Keywords: fixed point theory
zbMATH Keywords: Bézout theorem
zbMATH Keywords: Euler-Jacobi formula
zbMATH Keywords: multilinear analysis
zbMATH Keywords: syzygies
reviewed by: Pilar Rueda
MaRDI profile type: Publication
full work available at URL: dcdss.2013.6.1017
OpenAlex ID: W2320828795
DOI: 10.3934/DCDSS.2013.6.1017
description: scientific article; zbMATH DE number 6144761