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.
Similar content being viewed by others
References
Armand Borel,Density properties for certain subgroups of semi-simple groups without compact components, Ann. of Math. (2)72 (1960), 179–188.
Rufus Bowen,Weak mixing and unique ergodicity on homogeneous spaces, Israel J. Math.23 (1976), 267–273.
Gary Laison,A semigroup associated with an invariant measure on a transformation group, Math. Systems Theory8 (1975), 276–288.
Brian Marcus,Unique ergodicity of the horocycle flow; variable negative curvature case, preprint.
M. S. Raghunathan,Discrete Subgroups of Lie Groups, Springer-Verlag, New York, Heidelberg, Berlin, 1972.
A. Stepin,Dynamical systems on homogeneous spaces of semisimple Lie groups, Math. USSR-Izv.7 (1973), 1089–1104.
Author information
Authors and Affiliations
Additional information
The authors were partially supported by NSF grant MCS 75-05250.
Rights and permissions
About this article
Cite this article
Ellis, R., Perrizo, W. Unique ergodicity of flows on homogeneous spaces. Israel J. Math. 29, 276–284 (1978). https://doi.org/10.1007/BF02762015
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02762015