Descriptive properties of families of autohomeomorphisms of the unit interval
type: Item
instance of: scholarly article
title: Descriptive properties of families of autohomeomorphisms of the unit interval
zbMATH Open document ID: 1151.03024
author: Szymon Glab
published in: Journal of Mathematical Analysis and Applications
publication date: 2008-05-15T00:00:00Z
review text: Let \({\mathbb H}\subseteq C[0,1]\) denote the Polish space of all increasing autohomeomorphisms of \([0,1]\). An \(f\in {\mathbb H}\) is said to be strictly singular if \(f\) has no positive finite derivative at any point. Confirming a conjecture of \textit{S. Graf, R. D. Mauldin} and \textit{S. C. Williams} [``Random homeomorphisms'', Adv. Math. 60, 239--359 (1986; Zbl 0596.60005)], it is shown that SS\({\mathbb H}\), the set of all strictly singular autohomeomorphisms, is \(\Pi_1^1\)-complete. In addition, the following subsets of \({\mathbb H}\) are also shown to be \(\Pi_1^1\)-complete: SS\({\mathbb H}^+\) -- the set of \(f\in {\mathbb H}\) such that \(f\) has no finite positive right-hand derivative at any point, SS\({\mathbb H}^-\) -- the set of \(f\in {\mathbb H}\) such that \(f\) has no finite positive left-hand derivative at any point, \(\Delta_{<\infty}\) -- the set of \(f\in {\mathbb H}\) such that \(f\) has finite Dini derivative at every point, \(\Delta_{>0}\) -- the set of \(f\in {\mathbb H}\) such that \(f\) has positive Dini derivative at every point, and \(\Delta_{>0}\cap \Delta_{<\infty}\). These sets are also used to provide Borel-inseparable pairs of coanalytic sets.
reviewed by: Paul J. Szeptycki
Mathematics Subject Classification ID: 03E15
Mathematics Subject Classification ID: 28A05
Mathematics Subject Classification ID: 54H05
zbMATH DE Number: 5275666
zbMATH Keywords: singular autohomeomorphism
zbMATH Keywords: coanalytic set
zbMATH Keywords: \(\Pi_1^1\)-complete set
zbMATH Keywords: Dini derivative
zbMATH Keywords: Borel inseparable
MaRDI profile type: Publication
full work available at URL: j.jmaa.2008.01.068
OpenAlex ID: W2098636083
cites work: Some examples of Borel‐inseparable pairs of coanalylic sets
cites work: On the Dini derivates of a particular function
cites work: Random homeomorphisms
cites work: Q4323294
DOI: 10.1016/J.JMAA.2008.01.068
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description: scientific article; zbMATH DE number 5275666