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Minimal sets of homogeneous flows

Published online by Cambridge University Press:  19 September 2008

A. N. Starkov
A. N. Starkov
Affiliation:
Department of Mathematics, Moscow State University, 117234 Moscow, Russia
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Abstract

Generalizing Ratner's topological theorem for unipotent flows we obtain a sufficient condition for an orbit closure of a flow on a homogeneous space of finite volume to be algebraic. We give some applications of the result to studying birecurrent and minimal sets of homogeneous flows.

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Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 15 , Issue 2 , April 1995 , pp. 361 - 377
Copyright
Copyright © Cambridge University Press 1995

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References

REFERNCES

[A1]Auslander, L.. An exposition of the structure of solvmanifolds. Bull. Amer. Math. Soc. 79 (1973), 227–261.Google Scholar
[AB]Auslander, L. and Brezin, J.. Almost algebraic Lie algebras. J. Algebra 8 (1968), 195–313.CrossRefGoogle Scholar
[AG]Auslander, L. and Green, L.. G-induced flows. Amer. J. Math. 88 (1966), 43–60.CrossRefGoogle Scholar
[Bu]Burger, M.. Horocycle flow on geometrically finite surfaces. Duke Math. J. 61 (1990), 779–803.CrossRefGoogle Scholar
[D1]Dani, S. G.. On orbits of unipotent flows on homogeneous spaces. Ergod. Th. & Dynam. Sys. 4 (1984), 25–34.CrossRefGoogle Scholar
[DM]Dani, S. G. and Margulis, G. A.. Limit distributions of orbits of unipotent flows and values of quadratic forms. Adv. Sov. Math. 16 (1993), 91–137.Google Scholar
[H]Herman, M.. Minimal diffeomorphism of a positive entropy. Ergod. Th. & Dynam. Sys. 1 (1981), 65–76.CrossRefGoogle Scholar
[HM]Hochschild, G. and Mostow, G. D.. Representations and representative functions of Lie groups. Ann. Math. 66 (1957), 495–542.Google Scholar
[J]Johnson, R.. On almost periodic linear differential systems of Millionshchikov and Vinograd. J. Math. Anal. 85 (1982), 452–461.CrossRefGoogle Scholar
[KW]Katznelson, Y. and Weiss, B.. When all points are recurrent/generic. In: Katok, A. (ed.) Ergodic Theory and Dynamical Systems Vol. I, Proceedings, Special year, Maryland. 1979–1980. Birkhauser: Boston, 1981.Google Scholar
[Ma]Malcev, A. I.. Solvable Lie algebras. Izv. Akad. Nauk SSSR Ser. Mat. 9 (1945), 329–352.Google Scholar
[M1]Margulis, G. A.. Compactness of minimal closed invariant sets of actions of unipotent groups. Geom. Dedicala 37 (1991), 1–7.CrossRefGoogle Scholar
[M2]Margulis, G. A.. Dynamical and ergodic properties of subgroups actions on homogeneous spaces with applications to number theory. Proc. ICM-90, Kyoto, Japan. 193–215.Google Scholar
[R1]Ratner, M.. Raghunathan's toplogical conjecture and distributions of unipotent flows. Duke Math. J. 63 (1991), 235–280.Google Scholar
[R2]Ratner, M.. Strict measure rigidity for unipotent subgroups of solvable groups. Invent. Math. 101 (1990), 449–82.CrossRefGoogle Scholar
[R3]Ratner, M.. On Raghunathan's measure conjecture. Ann. Math. 134 (1991), 545–607.CrossRefGoogle Scholar
[Ra]Raghunathan, M. S.. Discrete Subgroups of Lie Groups. Springer: Berlin, 1972.CrossRefGoogle Scholar
[Re]Remage, R.. On minimal sets in the plane. Proc. Amer. Math. Soc. 13 (1962), 41–47.CrossRefGoogle Scholar
[S1]Starkov, A. N.. Orbit structures of homogeneous flows and Raghunathan's conjecture. Usp. Mat. Nauk. 45 (1990), 219–220.Google Scholar
[S2]Starkov, A. N.. Solvable homogeneous flows. Mat. Sbornik 134 (1987), 242–259.Google Scholar
[S3]Starkov, A. N.. An ergodic decomposition for homogeneous flows on finite volume homogeneous spaces. Mat. Sbornik 180 (1989), 1614–1633.Google Scholar
[S4]Starkov, A. N.. On mixing of higher degrees of homogeneous flows. Dokl. Ross. Akad. Nauk. 333:4, (1993), 28–31.Google Scholar
[S5]Starkov, A. N.. On the ergodicity criterion for G-induced flow. Usp. Mat. Nauk. 42 (1987), 197–198.Google Scholar
[W]Witte, D.. Zero-entropy affine maps on homogeneous spaces. Amer. J. Math. 109 (1987), 927–961.CrossRefGoogle Scholar
[Z]Zimmer, R.. Ergodic Theory and Semisimple Groups. Birkhauser: Boston, 1984.CrossRefGoogle Scholar