Cohomology of step functions under irrational rotations
type: Item
instance of: scholarly article
title: Cohomology of step functions under irrational rotations
zbMATH Open document ID: 0616.28010
DOI: 10.1007/BF02774084
author: Kathy D. Merrill
published in: Israel Journal of Mathematics
publication date: 1985-01-01T00:00:00Z
review text: Let \(X=[0,1]\) with addition mod 1 and let \(\theta\) be an irrational number in X. The author studies step functions f of X to the unit circle for which there exist constants \(\lambda\), \(| \lambda | =1\), and measurable functions g such that \((1)\quad g(x+\theta)=\lambda f(x)g(x)\) (that is, f is a multiple of a coboundary). Generalizing results of \textit{W. A. Veech} [Trans. Am. Math. Soc. 140, 1-33 (1969; Zbl 0201.056)], \textit{M. Stewart} [Acta Math. Acad. Sci. Hung. 37, 185-221 (1981; Zbl 0475.10040)] and \textit{K. Petersen} [Compos. Math. 26, 313-317 (1973; Zbl 0269.10030)], the author gives explicit formulas for solutions g of (1) for \(f=f_{t,s}=\exp (2\pi is\chi_{[1-t,1)})\) and \(t=k\theta \quad (mod 1).\) For a bigger set of t's it is shown that \(f_{t,s}\) is a multiple of a coboundary. For \(\theta\) with bounded partial quotients in its continued fraction expansion the author describes a family of step functions f (with finite number of points of discontinuity) for which (1) has no solution.
Mathematics Subject Classification ID: 28D05
Mathematics Subject Classification ID: 11K06
Mathematics Subject Classification ID: 11K38
Mathematics Subject Classification ID: 37A99
zbMATH DE Number: 3998140
zbMATH Keywords: multiplicative cocycle
zbMATH Keywords: irrational rotation
zbMATH Keywords: step functions
zbMATH Keywords: coboundary
reviewed by: Luchezar Stoyanov
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02774084
OpenAlex ID: W2065008577
description: scientific article; zbMATH DE number 3998140