References
[BBo]F. Bien, A. Borel, Sous-groupes épimorphiques de groupes algebriques lineaires I, C.R. Acad. Sci. Paris 315 (1992).
[BoP]A. Borel, G. Prasad, Values of isotropic quadratic forms atS-integral points, Composito Math. 83 (1992), 347–372.
[Bow]R. Bowen, Weak mixing and unique ergodicity on homogeneous spaces, Israel J. Math. 23 (1976), 267–273.
[D1]S.G. Dani, Invariant measures and minimal sets of horospherical flows Invent. Math. 64 (1981), 357–385.
[D2]S.G. Dani, On orbits of unipotent flows on homogeneous spaces, II Ergodic Theory and Dynamical Systems 6 (1986), 167–182.
[DM1]S.G. Dani, G.A. Margulis, Orbit closures of generic unipotent flows on homogeneous spaces ofS L (3, ℝ), Math. Ann. 286 (1990), 101–128.
[DM2]S.G. Dani, G.A. Margulis, Values of quadratic forms at primitive integral points, Invent. Math. 98 (1989), 405–425.
[DM3]S.G. Dani, G.A. Margulis, On limit distributions of orbits of unipotent flows and integral solutions of quadratic inequalities, C.R. Acad. Sci. Paris, Ser. I 314 (1992), 698–704.
[DSm]S.G. Dani, J. Smillie, Uniform distribution of horocycle orbits for Fuchsian groups, Duke Math. J. 5 (1984), 185–194.
[EPe]R. Ellis, W. Perrizo, Unique ergodicity of flows on homogeneous spaces, Israel J. Math. 29 (1978), 276–284.
[F1]H. Furstenberg, Strict ergodicity and transformations of the torus, Amer. J. Math. 83 (1961), 573–601.
[F2]H. Furstenberg, The unique ergodicity of the horocyle flow, In “Recent Advances in Topological Dynamics”, Lecture Notes in Math. 318, Springer, Berlin (1972), 95–115.
[H]G.A. Hedlund, Fuchsian groups and transitive horocycles, Duke Math. J. 2 (1936), 530–542.
[M1]G.A. Margulis, Discrete subgroups and ergodic theory, in “Number Theory, Trace Formula and Discrete Groups”, Symposium in Honor of A. Selberg, Oslo 1987, Academic Press (1989), 377–398.
[M2]G.A. Margulis, Dynamical and ergodic properties of subgroup actions on homogeneous spaces with applications to number theory, Proc. of ICM Kyoto (1990), 193–215.
[MT]G.A. Margulis, G.M. Tomanov, Measure rigidity for algebraic groups over local fields, C.R. Acad. Sci. Paris, t. 315, Serie I (1992), 1221–1226.
[Mo]S. Mozes, Epimorphic subgroups and invariant measures, Preprint.
[MoS]S. Mozes, N. Shah, On the space of ergodic measures of unipotent flows, preprint.
[P]W. Parry, Ergodic properties of affine transformations and flows on nilmanifolds, Amer. J. Math. 91 (1969), 757–771.
[R]M.S. Raghunathan, Discrete Subgroups of Lie Groups, Springer-Verlag (1972).
[Ra1]M. Ratner, Strict measure rigidity for unipotent subgroups of solvable groups, Invent. Math. 101 (1990), 449–482.
[Ra2]M. Ratner, On measure rigidity of unipotent subgroups of semisimple groups, Acta Math. 165 (1990), 229–309.
[Ra3]M. Ratner, On Raghunathan's measure conjecture, Ann. of Math. 134 (1991), 545–607.
[Ra4]M. Ratner, Raghunathan's topological conjecture and distributions of unipotent flows, Duke Math. J. 63 (1991), 235–280.
[Ra5]M. Ratner, Rigidity of horocycle flows, Ann. of Math. 115 (1982), 597–614.
[Ra6]M. Ratner, Raghunathan's Conjectures forSL(2,ℝ), Israel. J. Math. 80 (1992), 1–31.
[Ra7]M. Ratner, Raghunathan's conjectures forp-adic Lie groups, International Mathematics Research Notices, No. 5 (1993), 141–146.
[S]N. Shah, Uniformly distributed orbits of certain flows on homogeneous spaces, Math. Ann. 289 (1991), 315–334.
[V]W. Veech, Unique ergodicity of horospherical flows, Amer. J. Math. 99 (1977), 827–859.
[W1]D. Witte, Rigidity of some translations on homogeneous spaces, Invent. Math. 81 (1985), 1–27.
[W2]D. Witte, Measurable quotients of unipotent translations on homogeneous spaces, To appear in Transactions of AMS.
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Ratner, M. Invariant measures and orbit closures for unipotent actions on homogeneous spaces. Geometric and Functional Analysis 4, 236–257 (1994). https://doi.org/10.1007/BF01895839
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DOI: https://doi.org/10.1007/BF01895839