Inverse limits and full families
type: Item
instance of: scholarly article
title: Inverse limits and full families
zbMATH Open document ID: 1063.54031
author: Jonathon E. Fassett
published in: Topology and its Applications
publication date: 2004-08-19T00:00:00Z
review text: As first noticed by \textit{R. F. Williams} [Topology 6, 473--487 (1967; Zbl 0159.53702)] and \textit{L. Block} [Trans. Am. Math. Soc. 214, 403--413 (1975; Zbl 0293.57016)] there is a connection between inverse limits and attractors of dynamical systems. Different authors studied since the relationship between the dynamics of the bonding map and the topology of the resulting inverse limit. Many of these results are based on the tent or logistic familly of maps which are both well understood families of interval maps. The present paper gives results for inverse limits on the compact unit interval taking a single bonding map from a full family of one-parameter \(C^1\) unimodal maps \(f_\lambda\). The kneading theory and renormalization operator introduced by Feigenbaum are used in the proofs. It is shown that for \(\lambda\) such that the kneading sequence of \(f_\lambda\) is below that of \(f_{\lambda _\infty}\) where \(\lambda _\infty\) is the Feigenbaum value, the inverse limit with bonding map \(f_\lambda\) is hereditarily decomposable with topological \(\sin(\frac 1x)\)-curves as the dominant subcontinua. Approaching \(\lambda _\infty\) from above similar appearance of Brouwer-Janiszewski-Knaster indecomposable continuum is observed.
Mathematics Subject Classification ID: 54H20
Mathematics Subject Classification ID: 54F15
Mathematics Subject Classification ID: 37F25
zbMATH DE Number: 2092801
zbMATH Keywords: Inverse limits
zbMATH Keywords: Full families
zbMATH Keywords: Kneading theory
zbMATH Keywords: Renormalization
MaRDI profile type: Publication
full work available at URL: j.topol.2003.11.002
OpenAlex ID: W2042095284
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DOI: 10.1016/J.TOPOL.2003.11.002
description: scientific article; zbMATH DE number 2092801