Planar embeddings of inverse limit spaces of unimodal maps
type: Item
instance of: scholarly article
title: Planar embeddings of inverse limit spaces of unimodal maps
zbMATH Open document ID: 0954.54019
DOI: 10.1016/S0166-8641(98)00054-6
author: Henk Bruin
published in: Topology and its Applications
publication date: 2001-02-04T00:00:00Z
review text: A dynamical system \(f:X\to X\) can be represented by a homeomorphism \(\widehat{f}\) on a (compact) inverse limit space \(\widehat{X}=\{(\dots,x_{-1},x_0):f(x_{i})=x_{i+1}\}\) with the usual metric. For unimodal maps \(\widehat{X}\) may be embedded in \(\mathbb{R}^2\) [\textit{R. H. Bing}, Duke Math. J. 18, 653-663 (1951; Zbl 0043.16804); \textit{M. Misiurewicz}, Fundam. Math. 125, 23-40 (1985; 587.58032)]. The author proves that: If \(f\) is \(C^r\), \(r\geq 0\) then there exists a compact \(A\subset W\subset \mathbb{R}^2\), \(W\) a topological disk, and a map \(g:W\to W\) such that \(g(A)=A\), \((A,g)\) is topologically conjugate to \((\widehat{X},\widehat{f})\), \(g\) is Lipschitz on \(A\) and \(C^r\) on \(W\smallsetminus A\), \(A\) attracts some open neighbourhood of itself. Moreover, the embedding is sufficiently nice to preserve some measure theoretical properties of \(f\), in particular some pathological behaviour of physical (SBR)-measures. The proofs are mainly based on symbolic dynamics of \(f\) (kneading sequences).
reviewed by: Tomasz Nowicki
Mathematics Subject Classification ID: 54H20
Mathematics Subject Classification ID: 37B10
Mathematics Subject Classification ID: 37D45
zbMATH DE Number: 1340383
zbMATH Keywords: unimodal map
zbMATH Keywords: attractor
zbMATH Keywords: end-point
MaRDI profile type: Publication
full work available at URL: s0166-8641(98)00054-6
OpenAlex ID: W2057914782
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description: scientific article; zbMATH DE number 1340383