Hostname: page-component-7dd5485656-gs9qr Total loading time: 0 Render date: 2025-10-31T13:44:10.721Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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.

Information

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

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