Abstract
LetG be a unimodular Lie group, Γ a co-compact discrete subgroup ofG and ‘a’ a semisimple element ofG. LetT a be the mapgΓ →ag Γ:G/Γ →G/Γ. The following statements are pairwise equivalent: (1) (T a, G/Γ,θ) is weak-mixing. (2) (T a, G/Γ) is topologically weak-mixing. (3) (G u, G/Γ) is uniquely ergodic. (4) (G u, G/Γ,θ) is ergodic. (5) (G u, G/Γ) is point transitive. (6) (G u, G/Γ) is minimal. If in additionG is semisimple with finite center and no compact factors, then the statement “(T a, G/Γ,θ) is ergodic” may be added to the above list.
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The authors were partially supported by NSF grant MCS 75-05250.
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Ellis, R., Perrizo, W. Unique ergodicity of flows on homogeneous spaces. Israel J. Math. 29, 276–284 (1978). https://doi.org/10.1007/BF02762015
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DOI: https://doi.org/10.1007/BF02762015