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On uniformly distributed orbits of certain horocycle flows

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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Let

(where t ε ℝ) and let μ be the G-invariant probability measure on G/Γ. We show that if x is a non-periodic point of the flow given by the (ut)-action on G/Γ then the (ut)-orbit of x is uniformly distributed with respect to μ; that is, if Ω is an open subset whose boundary has zero measure, and l is the Lebesque measure on ℝ then, as T→∞, converges to μ(Ω).

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 2 , Issue 2 , June 1982 , pp. 139 - 158
Copyright
Copyright © Cambridge University Press 1982

References

REFERENCES

[1]Dani, S. G.. Invariant measures of horospherical flows on non-compact homogeneous spaces. Invent. Math. 47 (1978), 101–138.Google 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.Google Scholar
[4]Dani, S. G. & Raghavan, S.. Orbits of Euclidean frames under discrete linear groups. Israeli. Math. 36 (1980), 300–320.CrossRefGoogle Scholar
[5]Furstenberg, H.. The unique ergodicity of the horocycle flow, Recent Advances in Topological Dynamics (ed. Beck, A.), pp. 95–115. Springer: Berlin, 1972.Google Scholar
[6]Katznelson, Y. & Weiss, B.. When all points are recurrent/generic. In Ergodic Theory and Dynamical Systems I, pp. 195–210 (ed. Katok, A.), Proceedings, Special year, Maryland 1979–80. Birkhauser: Boston, 1981.Google Scholar
[7]Moore, C. C.. Ergodicity of flows on homogeneous spaces. Amer. J. Math. 88 (1966), 154–178.CrossRefGoogle Scholar
[8]Raghunathan, M. S.. Discrete Subgroups of Lie Groups. Springer: Berlin, 1972.CrossRefGoogle Scholar
[9]Siegel, C. L.. A mean value theorem in geometry of numbers. Ann. Math. 46 (1945), 340–347.Google Scholar
[10]Veech, W. A.. Unique ergodicity of horospherical flows. Amer. J. Math. 99 (1977), 827–859.Google Scholar