Ergodicity of certain cylinder flows
type: Item
instance of: scholarly article
title: Ergodicity of certain cylinder flows
zbMATH Open document ID: 0764.28011
DOI: 10.1007/BF02782848
author: D. A. Pask
published in: Israel Journal of Mathematics
publication date: 1992-12-09T00:00:00Z
review text: Let \(T\) be an ergodic transformation on the torus \(\mathbb{T}\) that preserves the normalized Haar measure \(\lambda\). Let \(f:\mathbb{T}\to\mathbb{R}\) be a Borel measurable function. Suppose further that the closed subgroup of \(\mathbb{R}\) generated by the values of \(f\) equals \(\mathbb{R}\). Then it makes sense to define the skew product extension \(T_ f\) of \(T\) by \(f\) on the product space \(\mathbb{T}\times\mathbb{R}\): \[ T_ f:\mathbb{T}\times\mathbb{R}\to\mathbb{T}\times\mathbb{R},\quad T_ f(x,y):=(Tx,y+f(x)). \] Denote the measure theoretic product of \(\lambda\) with Lebesgue measure \(\lambda_ 1\) on \(\mathbb{R}\) by \(\mu\). Ergodicity of the transformation \(T_ f\) with respect to \(\mu\) obviously depends on the interplay between \(f\) and \(T\). A necessary condition for ergodicity is that \(f\) has integral zero. Define \(f_ n(x):=\sum^{n-1}_{k=0}f(T^ kx)\) \((x\in\mathbb{T},n\in\mathbb{N})\). Then the \(n\)-th iterate of \(T_ f\) has the form \(T^ n_ f(x,y)=(T^ nx,y+f_ n(x))\). Boundedness of the sequence \(f_ n\) prohibits \(T_ f\) from being ergodic. In this case, \(f\) is a coboundary. For this notion and a survey of related results see \textit{P. Liardet} [Compos. Math. 61, 267-293 (1987; Zbl 0619.10053)]. Let \(T\) be the irrational rotation of \(\mathbb{T}\) defined by \(\alpha\), \(Tx:=x+\alpha\;\bmod 1\). The author studies the set \[ {\mathcal E}(f):=\{\alpha\text{ irrational : }T_ f\text{ is ergodic}\} \] for certain classes of differentiable functions \(f\). If \(f\) is in \(C^ 1[0,1]\), with \(f(0)\neq f(1)\), then, by \textit{P. Hellekalek} and \textit{G. Larcher} [Isr. J. Math. 54, 301-306 (1986; Zbl 0609.28007)], \({\mathcal E}(f)\) is equal to the set of all irrational numbers. This result was generalized by the author [Isr. J. Math. 69, No. 1, 65-74 (1990; Zbl 0703.28009)] to piecewise absolutely continuous functions \(f\) with Riemann-integrable derivative \(f'\) with nonzero integral. In the case \(f\in C^ 1[0,1]\), \(…
reviewed by: Peter Hellekalek
Mathematics Subject Classification ID: 28D05
Mathematics Subject Classification ID: 28D10
Mathematics Subject Classification ID: 11K38
zbMATH DE Number: 78198
zbMATH Keywords: ergodicity
zbMATH Keywords: skew product
zbMATH Keywords: coboundary
zbMATH Keywords: irrational rotation
zbMATH Keywords: von Neumann-Kakutani adding machine transformations
zbMATH Keywords: essential values of cocycles
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02782848
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description: scientific article; zbMATH DE number 78198