By a result of R. J. Zimmer [J. Funct. Anal. 27, 350-372 (1978; Zbl 0391.28011)] every group action representable as an image of a cocycle of some amenable action (in particular of a single automorphism) is itself amenable. The converse assertion was proved earlier only for free actions [A. Connes, J. Feldman and B. Weiss, Ergodic Theory Dyn. Syst. 1, 431-450 (1981; Zbl 0491.28018) and J. Feldman, P. Hahn and C. C. Moore, Adv. Math. 28, 186-230 (1978; Zbl 0392.28023), Corollary 7.9] and is here given in full scope (with a sketch of the proofs).
Theorem 1. Suppose that the locally compact separable (LCS) group G acts ergodically and amenably on the Lebesgue space (S,\(\mu\)). Then there exists a cocycle of an ergodic automorphism of any previously specified type whose image is this action.
Corollary. Let G be an LCS group, and let an amenable ergodic action of the group \(G\times {\mathbb{R}}\) be given. Then there exists a cocycle \(\alpha\) of an ergodic automorphism T with values in G such that the given action of \(G\times {\mathbb{R}}\) is the image of the cocycle \(\alpha\times r\), where r is the Radon-Nikodým cocycle of T.
Theorem 2. Suppose that the LCS group G acts ergodically and amenably on the space (S,\(\mu\)). Then for a.e. \(s\in S\) the isotropy subgroup \(G_ s\) of the point s is amenable.
Theorem 3. Suppose that an LCS group G acts ergodically on the space (X,\(\mu\)) in such a way that the corresponding equivalence relation \(R_ G\) on X is amenable, and for a.e. \(\chi\in X\) the isotropy subgroup \(G_{\chi}\) of the point \(\chi\) is amenable. Then this action is amenable.
A closed related result was obtained by A. L. Fedorov [Krieger’s theorem for cocycles, Preprint, Tashkent (1985; R. Zh. Mat. 1985, 6Б919)].