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On orbits of unipotent flows on homogeneous spaces

Published online by Cambridge University Press:  19 September 2008

S. G. Dani
S. G. Dani
Affiliation:
School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay 400005, India
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Abstract

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Let G be a connected Lie group and let Γ be a lattice in G (not necessarily co-compact). We show that if (ut) is a unipotent one-parameter subgroup of G then every ergodic invariant (locally finite) measure of the action of (ut) on G/Γ is finite. For ‘arithmetic lattices’ this was proved in [2]. The present generalization is obtained by studying the ‘frequency of visiting compact subsets’ for unbounded orbits of such flows in the special case where G is a connected semi-simple Lie group of ℝ-rank 1 and Γ is any (not necessarily arithmetic) lattice in G.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 4 , Issue 1 , March 1984 , pp. 25 - 34
Copyright
Copyright © Cambridge University Press 1984

References

REFERENCES

[1]Dani, S. G.. Invariant measures of horospherical flows on non-compact homogeneous spaces. Invent. Math. 47 (1978), 101–138.CrossRefGoogle Scholar
[2]Dani, S. G.. On invariant measures, minimal sets and a lemma of Margulis. Invent. Math. 51 (1979), 239–260.CrossRefGoogle Scholar
[3]Dani, S. G.. Invariant measures and minimal sets of horospherical flows. Invent. Math. 64 (1981), 357–385.CrossRefGoogle Scholar
[4]Dani, S. G. & Smillie, John. Uniform distribution of horocycle orbits for Fuchsian groups. Duke Math. J. 51 (1984), 185–194.CrossRefGoogle Scholar
[5]Garland, H. & Raghunathan, M. S.. Fundamental domains for lattices in ℝ-rank 1 semisimple Lie groups. Ann. of Math. 92 (1970), 279–326.CrossRefGoogle Scholar
[6]Margulis, G. A.. On the action of unipotent groups in the space of lattices. Proc. of the Summer School on Group Representations, Budapest, Bolyai Janos Math. Soc., pp. 365–370, 1971.Google Scholar
[7]Raghunathan, M. S.. Discrete Subgroups of Lie Groups. Springer-Verlag: Berlin-Heidelberg-New York (1972).CrossRefGoogle Scholar