Minimal sets of periods for torus maps
type: Item
instance of: scholarly article
title: Minimal sets of periods for torus maps
zbMATH Open document ID: 0965.37019
author: Jaume Llibre
author: Boju Jiang
published in: Discrete and Continuous Dynamical Systems
publication date: 2000-08-16T00:00:00Z
review text: Let \(f:X\to X\) be a self-map of a compact connected polyhedron \(X\), and \(n\) be a natural number. For \(\text{Fix} (f)\) and \(P_n(f)\) we denote \[ \text{Fix}(f)= \{x\in X\mid x=f(x)\}, \quad P_n(f)= \{x\in X\mid x= f^n(x), \text{ but }x\neq f^k(x),\;k< n\}. \] Denote by \(\text{Per}(f)= \{n\in N\mid P_n(f)\neq 0\}\) and by \(M \operatorname {Per}(f)= \bigcap \operatorname {Per}(g)\), where the intersection is over all \(g\) which are homotopy equivalent to \(f\). The present paper deals with the problem of determining the set of periods of the periodic orbits of a map given the homotopy class of the map. The authors characterize the set \(M\operatorname {Per}(f)\) in terms of Nielsen numbers of the iterates of \(f\). Furthermore the authors classify all the sets \(M\operatorname {Per}(f)\) for self-maps of the 3D torus.
Mathematics Subject Classification ID: 37C25
Mathematics Subject Classification ID: 37E99
Mathematics Subject Classification ID: 37C30
zbMATH DE Number: 1492521
zbMATH Keywords: minimal set
zbMATH Keywords: periodic orbit
zbMATH Keywords: homotopy class
zbMATH Keywords: Nielsen numbers
reviewed by: Messoud Efendiev
MaRDI profile type: Publication
full work available at URL: dcds.1998.4.301
OpenAlex ID: W2013342692
DOI: 10.3934/DCDS.1998.4.301
description: scientific article; zbMATH DE number 1492521