Dynamics of typical Baire-1 functions on the interval
type: Item
instance of: scholarly article
title: Dynamics of typical Baire-1 functions on the interval
zbMATH Open document ID: 1379.54034
published in: Journal of Applied Analysis
publication date: 2017-12-04T00:00:00Z
review text: Let \(B\) denote the set of Baire class 1 functions \([0,1]\to[0,1]\) (that is, pointwise limits of continuous functions), and write as usual \(\omega(x,f)\) for the set of cluster points of the forward orbit of \(x\) under iteration of \(f\in B\). The two main theorems are these. First, there is a residual set \(S\subset B\) with the property that for any function \(f\in S\) the set \(\omega(x,f)\) is contained in the set of continuity points of \(f\) and on \(\omega(x,f)\) the map is an \(\infty\)-adic adding machine, for every \(x\in[0,1]\). Second, there is a residual set \(S'\subset B\) with the property that for any \(f\in S'\) and \(\epsilon>0\) there is some \(M\) for which \(f^m([0,1])\) is contained in the \(\epsilon\)-neighbourhood of \(\bigcup_{x\in[0,1]}\omega(x,f)\) for \(m>M\). Somewhat surprisingly, these results are similar to similar statements known to hold in the more familiar setting of iteration of continuous maps of the interval (see, for example, the paper of \textit{H. Lehning} [Proc. Am. Math. Soc. 123, No. 6, 1703--1707 (1995; Zbl 0843.58077)] and its references).
reviewed by: Thomas B. Ward
Mathematics Subject Classification ID: 54H20
Mathematics Subject Classification ID: 37B99
Mathematics Subject Classification ID: 26A18
zbMATH DE Number: 6815246
zbMATH Keywords: Baire-1 function
zbMATH Keywords: \(\omega\)-limit set
zbMATH Keywords: odometer
zbMATH Keywords: generic
author: Timothy H. Steele
MaRDI profile type: Publication
full work available at URL: jaa-2017-0009
OpenAlex ID: W2768682825
cites work: Dynamics of Typical Continuous Functions
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DOI: 10.1515/JAA-2017-0009
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description: scientific article; zbMATH DE number 6815246