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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
##### Keywords:
continuous cocycles; Lie groups
##### Citations:
Zbl 0414.28024; Zbl 0625.28008
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