Gutschera, K. Robert Ergodic elements for actions of Lie groups. (English) Zbl 0858.58028 Ergodic Theory Dyn. Syst. 16, No. 4, 703-717 (1996). 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) Cited in 2 Documents 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 Keywords:ergodic groups; ergodic action; connected Lie group; irreducible representations; semisimple groups; solvable groups; Levi decomposition PDFBibTeX XMLCite \textit{K. R. Gutschera}, Ergodic Theory Dyn. Syst. 16, No. 4, 703--717 (1996; Zbl 0858.58028) Full Text: DOI References: [1] Dani, Dense Orbits of Horospherical Flows, Dynamical Systems and Ergodic Theory 23 (1989) · Zbl 0701.58050 [2] DOI: 10.1016/0022-1236(79)90078-8 · Zbl 0404.22015 · doi:10.1016/0022-1236(79)90078-8 [3] Auslander, Mem. Amer. Math. Soc. 62 pp none– (1966) [4] DOI: 10.1090/S0002-9904-1973-13134-9 · Zbl 0265.22016 · doi:10.1090/S0002-9904-1973-13134-9 [5] Zimmer, Ergodic Theory and Semisimple Groups (1984) · doi:10.1007/978-1-4684-9488-4 [6] DOI: 10.2307/2374057 · Zbl 0507.22011 · doi:10.2307/2374057 [7] DOI: 10.2307/2944357 · Zbl 0763.28012 · doi:10.2307/2944357 [8] Pugh, Compositio Mathematica 23 pp 115– (1971) [9] Ornstein, J. Analyse Math. 48 pp 1– (1987) [10] DOI: 10.2307/2373052 · Zbl 0148.37902 · doi:10.2307/2373052 [11] Moore, Pacific J. Math. 86 pp 155– (1980) · Zbl 0446.22014 · doi:10.2140/pjm.1980.86.155 [12] Lipsman, Group Representations. Lecture Notes in Mathematics 388 (1974) [13] DOI: 10.1070/RM1962v017n04ABEH004118 · Zbl 0106.25001 · doi:10.1070/RM1962v017n04ABEH004118 [14] Jacobson, Lie Algebras (1979) [15] DOI: 10.2307/2374105 · Zbl 0506.22008 · doi:10.2307/2374105 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.