Ergodic affine maps of locally compact groups
It is proved that if a locally compact group G admits a continuous affine map which is ergodic with respect to the Haar measure, then it is either compact or discrete. (A map T is said to be affine if there exists \(a\in G\) and a group automorphism f such that \(T(g)=a f(g)\) for all \(g\in G).\) For (continuous) group automorphisms the result was proved by \textit{N. Aoki} recently [Topology Appl. 20, 1-15 (1985; Zbl 0568.28013)], in response to the question raised in \textit{P. R. Halmos}' book 'Lectures on ergodic theory' (1956; Zbl 0073.093); in this case, in fact, G has to be compact. For affine maps of connected Lie groups the results was proved earlier by the reviewer [cf. J. Lond. Math. Soc., II. Ser. 25, 241-245 (1982; Zbl 0451.22004)]. The present authors use the reviewer's result and various results and techniques from Aoki's paper mentioned above, to conclude the general result.
scientific article; zbMATH DE number 3927247