Inverse limits on [0,1] using logistic bonding maps
type: Item
instance of: scholarly article
title: Inverse limits on [0,1] using logistic bonding maps
zbMATH Open document ID: 0859.54030
DOI: 10.1016/0166-8641(96)00025-9
author: W. T. Ingram
author: Marcy Barge
published in: Topology and its Applications
publication date: 1997-04-07T00:00:00Z
review text: Let \(f_\lambda: [0,1]\to [0,1]\) be the logistic map defined by \(f_\lambda(x)= 4\lambda x(1-x)\), for \(0\leq\lambda\leq 1\), and let \(M_\lambda\) denote the inverse limit continuum obtained by using \(f_\lambda\) for each one-step bonding map. The authors investigate the topological structure of \(M_\lambda\) and relate this to known dynamics for the logistic family of maps. For \(0\leq\lambda\leq 1/4\), \(M_\lambda\) is a singleton. For \(1/4<\lambda< \lambda_c\) (\(\approx 0.89249\), the Feigenbaum limit), \(M_\lambda\) is hereditarily decomposable, consisting of a ray limiting to collections of \(\sin(1/x)\)-curves. For \(\lambda= \lambda_c\), \(M_\lambda\) is hereditarily decomposable with only three types of nondegenerate subcontinua: arcs, copies of \(M_\lambda\), and unions of two copies of \(M_\lambda\) intersecting at a common endpoint of a ray. For \(\lambda> \lambda_c\), \(M_\lambda\) always contains a nondegenerate indecomposable continuum. The types and numbers of distinct nondegenerate indecomposable subcontinua present is investigated in terms of the periodic structure of \(f_\lambda\). Throughout, the paper is very well illustrated. While not providing a complete description of \(M_\lambda\) for each parameter value of \(\lambda\), it provides a very useful summary of the close tie between topology and dynamics for this basic family of maps.
reviewed by: Wayne Lewis
Mathematics Subject Classification ID: 54H20
Mathematics Subject Classification ID: 54F15
Mathematics Subject Classification ID: 37E99
Mathematics Subject Classification ID: 37B99
zbMATH DE Number: 937149
zbMATH Keywords: logistic map
zbMATH Keywords: inverse limit
zbMATH Keywords: indecomposable continuum
MaRDI profile type: Publication
full work available at URL: 0166-8641(96)00025-9
OpenAlex ID: W2092972143
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description: scientific article; zbMATH DE number 937149