On circle-valued cocycles of an ergodic measure preserving transformation
type: Item
instance of: scholarly article
title: On circle-valued cocycles of an ergodic measure preserving transformation
zbMATH Open document ID: 0652.28004
DOI: 10.1007/BF02776300
published in: Israel Journal of Mathematics
publication date: 1988-01-01T00:00:00Z
review text: Let \(\tau\) be an invertible ergodic measure preserving transformation of a probability space (X,\(\mu)\), and let f be a measurable function from X into the circle \({\mathbb{T}}\). If there is a measurable g: \(X\to {\mathbb{T}}\) and a scalar \(\lambda\), \(| \lambda | =1\), with \(f(x)=\lambda g(x)/g(\tau (x))\) for \(\mu\) almost all x, then f is called to be a projective coboundary of \(\tau\). The paper under review contains necessary and sufficient conditions for a measurable f: \(X\to {\mathbb{T}}\) to be a projective coboundary of \(\tau\). It is proved, in particular, that if there exists a sequence \(\{a_ n\}\) of complex numbers of modulus \(\leq 1\) such that \[ (1/N)\sum^{N- 1}_{n=0}a_ n\int f^{(n)}(x) d\mu (x) \] does not tend to 0, then f is a projective coboundary of \(\tau\). Here \(f^{(n)}(x)=f(x)f(\tau (x))...f(\tau^{n-1}(x)),\quad f^{(0)}(x)=1.\) Two applications are given.
Mathematics Subject Classification ID: 28D05
zbMATH DE Number: 4063450
zbMATH Keywords: cocycle
zbMATH Keywords: invertible ergodic measure preserving transformation
zbMATH Keywords: projective coboundary
author: Lawrence W. Baggett
reviewed by: Luchezar Stoyanov
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 4063450