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On orbits of unipotent flows on homogeneous spaces, II

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 400 005, India
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Abstract

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We show that if (ut) is a one-parameter subgroup of SL (n, ℝ) consisting of unipotent matrices, then for any ε > 0 there exists a compact subset K of SL(n, ℝ)/SL(n, ℤ) such that the following holds: for any g ∈ SL(n, ℝ) either m({t ∈ [0, T] | utg SL (n, ℤ) ∈ K}) > (1 – ε)T for all large T (m being the Lebesgue measure) or there exists a non-trivial (g−1utg)-invariant subspace defined by rational equations.

Similar results are deduced for orbits of unipotent flows on other homogeneous spaces. We also conclude that if G is a connected semisimple Lie group and Γ is a lattice in G then there exists a compact subset D of G such that for any closed connected unipotent subgroup U, which is not contained in any proper closed subgroup of G, we have G = DΓ U. The decomposition is applied to get results on Diophantine approximation.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 6 , Issue 2 , June 1986 , pp. 167 - 182
Copyright
Copyright © Cambridge University Press 1986

References

REFERENCES

[1]Borel, A.. Introduction aux Groupes Arithmétiques. Publ. de l'lnst. Math, de l'univ. de Strasbourg XV. Hermann, Paris, 1969.Google Scholar
[2]Chevalley, C.. Theorie des Groupes de Lie. Publ. de l'lnst. de Math, de l'univ. de Nancago I et IV. Hermann, Paris, 1968.Google Scholar
[3]Dani, S. G.. On invariant measures, minimal sets and a lemma of Margulis. Invent. Math. 51 (1979), 239–260.Google Scholar
[4]Dani, S. G.. A simple proof of Borel's density theorem. Math. Z. 174 (1980), 81–94.Google Scholar
[5]Dani, S. G.. Invariant measures and minimal sets of horospherical flows. Invent. Math. 64 (1981), 357–385.Google Scholar
[6]Dani, S. G.. On orbits of unipotent flows on homogeneous spaces. Ergod. Th. & Dynam. Sys. 4 (1984), 25–34.Google Scholar
[7]Dani, S. G.. Orbits of horospherical flows. To appear in Duke J. Math. 53 (1986).Google Scholar
[8]Dani, S. G. & Raghavan, S.. Orbits of Euclidean frames under discrete linear groups. Israel J. Math. 36 (1980), 300–320.Google Scholar
[9]Jacobson, N.. Lie Algebras. Interscience Publ. New York-London-Sydney, 1962Google Scholar
[10]Margulis, G. A.. On the action of unipotent groups in the space of lattices. In Proc. of the Summer School on Group-Representations. Bolyai Janos Math. Soc. pp. 365–370, Budapest, 1971.Google Scholar
[11]Margulis, G. A.. Arithmetic properties of discrete subgroups. Uspehi Mat. Nauk 29:1(1974) 49–98=Google Scholar
Russian Math. Surveys 29:1 (1974), 107–156.Google Scholar
[12]Raghunathan, M. S.. Discrete Subgroups of Lie Groups. Springer: Berlin-Heidelberg-New York, 1972.Google Scholar