Find out how to access previewonly content
Limit distributions of expanding translates of certain orbits on homogeneous spaces
 Nimish A. Shah
 … show all 1 hide
Rent the article at a discount
Rent now* Final gross prices may vary according to local VAT.
Get AccessAbstract
LetL be a Lie group and λ a lattice inL. SupposeG is a noncompact simple Lie group realized as a Lie subgroup ofL and\(\overline {GA} = L\). LetaεG be such that Ada is semisimple and not contained in a compact subgroup of Aut(Lie(G)). Consider the expanding horospherical subgroup ofG associated toa defined as U^{+} ={gεG:a ^{−n} ga^{n}} →e as n → ∞. Let Ω be a nonempty open subset ofU ^{+} andn _{i} → ∞ be any sequence. It is showed that\(\overline { \cup _{i = 1}^\infty a^n \Omega \Lambda } = L\). A stronger measure theoretic formulation of this result is also obtained. Among other applications of the above result, we describeGequivariant topological factors of L/gl × G/P, where the real rank ofG is greater than 1,P is a parabolic subgroup ofG andG acts diagonally. We also describe equivariant topological factors of unipotent flows on finite volume homogeneous spaces of Lie groups.
References
[D1]
Dani S G, Invariant measures of horospherical flows on noncompact homogeneous spaces,Invent. Math.
47 (1978) 101–138MATHCrossRefMathSciNet
[D2]
Dani S G, On orbits of unipotent flows on homogeneous spaces,Ergod. Th. Dynam. Sys.
4 (1980) 25–34MathSciNet
[D3]
Dani S G, Continuous equivariant images of latticeactions on boundaries,Ann. Math.
119 (1984) 111–119CrossRefMathSciNet
[DM1]
Dani S G and Margulis G A, Orbit closures of generic unipotent flows on homogeneous spaces of SL(3, ℝ),Math. Ann.
286 (1990) 101–128MATHCrossRefMathSciNet
[DM2]
Dani S G and Margulis G A, Asymptotic behaviour of trajectories of unipotent flows on homogeneous spaces,Proc. Indian Acad. Sci.
101 (1991) 1–17MATHMathSciNetCrossRef
[DM3]
Dani S G and Margulis G A, Limit distributions of orbits of unipotent flows and values of quadratic forms,Advances in Soviet Math.
16 (1993) 91–137MathSciNet
[DR]
Dani S G and Raghavan S, Orbits of Euclidean frames under discrete linear groups,Isr. J. Math.
36 (1980) 300–320MATHCrossRefMathSciNet
[DRS]
Duke W, Rudnick Z and Sarnak P, Density of integer points on affine symmetric varieties,Duke Math J.
71(1993) 143–180MATHCrossRefMathSciNet
[EM]
Eskin A and McMullen C, Mixing, counting and equidistribution in Lie groups,Duke Math. J.
71 (1993)181–209MATHCrossRefMathSciNet
[EMS1]
Eskin A, Mozes S and Shah N A, Nondivergence of translates of certain algebraic measures. Preprint
[EMS2]
Eskin A, Mozes S and Shah N A, Unipotent flows and counting lattice points on homogeneous varieties.Ann. Math. (to appear)
[GR]
Garland H and Raghunathan M S, Fundamental domains for lattices in ℝrank 1 semisimple Lie groups,Ann. Math.
92 (1970) 279–326CrossRefMathSciNet
[Ml]
Margulis G A, Quotient groups of discrete subgroups and measure theory, Funct.Anal. Appl.
12 (1978) 295–305MATHMathSciNet
[M2]
Margulis G A, Arithmeticity of irreducible lattices in semisimple groups of rank greater than 1,Invent Math.
76(1984) 93–120MATHCrossRefMathSciNet
[MS]
Mozes S and Shah N A, On the space of ergodic invariant measures of unipotent flows,Ergod. Th. Dynam. Sys.
15 (1978) 149–159MathSciNet
[R]
Raghunathan M S,Discrete subgroups of Lie groups. (1972) (Springer: Berlin Heidelberg New York)MATH
[Ra1]
[Ra2]
Ratner M, Raghunathan’s topological conjecture and distributions of unipotent flows.Duke Math. J.
63 (1991) 235–280MATHCrossRefMathSciNet
[S]
Schlichtkrull H,Hyperfunctions and Harmonic Analysis on Symmetric Spaces (1984) (Birkhauser: Boston)MATH
[Sh1]
Shah N A, Uniformly distributed orbits of certain flows on homogeneous spaces.Math. Ann.
289 (1991) 315–334MATHCrossRefMathSciNet
[Sh2]
Shah N A, Limit distributions of polynomial trajectories on homogeneous spaces,Duke Math. J.
75 (1994) 711–732MATHCrossRefMathSciNet
[St]
Stuck G, Minimal actions of semisimple groups: InWorkshop on Lie groups, Ergodic theory, and Geometry and problems and Geometric rigidity, April 13–17, 1992, Berkeley. MSRI preprint (August 1992).
[SZ]
Stuck G and Zimmer R J, Stabilizers of ergodic actions of higher rank semisimple groups,Ann. Math.
139 (1994) 723–747MATHCrossRefMathSciNet
[W]
Witte D, Measurable quotients of unipotent translations on homogeneous spaces,Trans. Am. Math. Soc.
345 (1994) 577–594MATHCrossRefMathSciNet
[Z1]
Zimmer R J, Ergodic theory, semisimple groups, and foliations by manifolds of negative curvature.I.H.E.S. Publ. Math.
55 (1982) 37–62MATHMathSciNet
 Title
 Limit distributions of expanding translates of certain orbits on homogeneous spaces
 Journal

Proceedings of the Indian Academy of Sciences  Mathematical Sciences
Volume 106, Issue 2 , pp 105125
 Cover Date
 19960501
 DOI
 10.1007/BF02837164
 Print ISSN
 03700089
 Online ISSN
 09737685
 Publisher
 Springer India
 Additional Links
 Topics
 Keywords

 Limit distributions
 unipotent flow
 horospherical patches
 symmetric subgroups
 continuous equivariant factors
 Authors

 Nimish A. Shah ^{(1)}
 Author Affiliations

 1. School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, 400 005, Bombay, India