Fixed point index and decompositions of isolated invariant compacta
type: Item
instance of: scholarly article
title: Fixed point index and decompositions of isolated invariant compacta
zbMATH Open document ID: 1054.54031
published in: Topology and its Applications
publication date: 2004-08-20T00:00:00Z
review text: Let \(U\subset \mathbb{R}^2\) be an open set and let \(f:U\rightarrow f(U)\subset\mathbb{R}^2\) be a homeomorphism. Let \(M=M_1\cup\dots\cup M_r\subset U\) be a disjoint union of discs that isolates the invariant compactum \(K\), i.e., \(K=\{x\in M\mid \exists\, \sigma:\mathbb{Z}\rightarrow M\) such that \(\sigma(0)=x\) and \(f(\sigma(i))=\sigma(i+1)\) for all \(i\in\mathbb{Z}\}\). The aim of this paper is to provide techniques to study the dynamics of \(f\) in \(K\) and, using fixed point index theory, to detect the periodic orbits of \(f\) in \(K\) following a given itinerary \(\tau\). Moreover, in some cases containing classical and important discrete dynamical systems the author computes the fixed point index of every iteration of \(f\) in small neighborhoods of the detected periodic points.
reviewed by: Mircea Balaj
Mathematics Subject Classification ID: 54H25
Mathematics Subject Classification ID: 54H20
zbMATH DE Number: 2093561
zbMATH Keywords: fixed point index
zbMATH Keywords: Conley index
zbMATH Keywords: filtration pairs
author: Francisco R. Ruiz del Portal
author: José Miguel Salazar
MaRDI profile type: Publication
full work available at URL: j.topol.2003.12.013
OpenAlex ID: W2019185216
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DOI: 10.1016/J.TOPOL.2003.12.013
description: scientific article; zbMATH DE number 2093561