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On the existence of cohomologous continuous cocycles for cocycles with values in some Lie groups. (English) Zbl 0984.37002
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 A. V. Kochergin [Dokl. Akad. Nauk. SSSR 231, 795-798 (1976; Zbl 0414.28024)] and D. J. Rudolph [Ergodic Theory Dyn. Sys. 6, 583-599 (1986; Zbl 0625.28008)].
MSC:
37A20 Algebraic ergodic theory, cocycles, orbit equivalence, ergodic equivalence relations
28D15 General groups of measure-preserving transformations
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