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Ergodic elements for actions of Lie groups. (English) Zbl 0858.58028

Let \(G\) be a connected Lie group acting ergodically on a space \(S\) with a finite invariant measure \(\mu\). It is proved in this paper that for most types of groups there exists in \(G\) a single element which acts ergodically on \(S\), but not for compact non-Abelian groups or for certain types of solvable groups. More precisely, the following holds: \(G\) always has ergodic elements if and only if \(G\) does not project onto a compact simple group or the isometry group of the plane.
The proof relies on using the theory of irreducible representations to examine the action of \(G\) on the Hilbert space of \(L^2 (S,\mu)\). The cases of semisimple groups and solvable groups are studied in detail and the conclusion follows from the Levi decomposition of \(G\).
Reviewer: J.Lacroix (Paris)

MSC:

37A99 Ergodic theory
28D15 General groups of measure-preserving transformations
22E25 Nilpotent and solvable Lie groups
22E10 General properties and structure of complex Lie groups
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