On the existence of cohomologous continuous cocycles for cocycles with values in some Lie groups
type: Item
instance of: scholarly article
title: On the existence of cohomologous continuous cocycles for cocycles with values in some Lie groups
zbMATH Open document ID: 0984.37002
DOI: 10.1016/S0246-0203(00)00124-2
author: Jean-Marc Derrien
published in: Annales de l'Institut Henri Poincaré. Probabilités et Statistiques
publication date: 2001-01-03T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/77660
review text: Let \(X\) be a compact metric space with a non-atomic Borel probability \(\mu\), and \(T:X\to X\) an ergodic invertible \(\mu\)-preserving transformation. In this paper the following rather general result is proved for the cohomology of measurable cocycles. Let \(G\) be a connected Lie group with a Riemannian metric invariant under right and left translations (this is a condition on \(\text{Ad}(G)\), satisfied for example when \(G\) is compact or abelian). Then any cocycle \(\varphi:X\to G\) that is integrable (in the sense that the geodesic distance from the identity element is integrable over \(X\)) has an associated continuous cocycle \(\Phi:X\to G\) and a measurable transfer function \(\psi:X\to G\) with \(\Phi(x)= \psi(T^{-1}x)\varphi(x)\psi(x)\) a.e. This generalizes results due to \textit{A. V. Kochergin} [Dokl. Akad. Nauk. SSSR 231, 795-798 (1976; Zbl 0414.28024)] and \textit{D. J. Rudolph} [Ergodic Theory Dyn. Sys. 6, 583-599 (1986; Zbl 0625.28008)].
Mathematics Subject Classification ID: 37A20
Mathematics Subject Classification ID: 28D15
zbMATH DE Number: 1501487
zbMATH Keywords: continuous cocycles
zbMATH Keywords: Lie groups
reviewed by: Thomas B. Ward
MaRDI profile type: Publication
Wikidata QID: Q115339489
OpenAlex ID: W2012898166
description: scientific article; zbMATH DE number 1501487