On orbits of unipotent flows on homogeneous spaces

Dani, S. G.

doi:10.1017/s0143385700002248

Cited by 67 publications

(75 citation statements)

References 7 publications

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“…This phenomenon of non-divergence was quantified and further refined by Dani, Kleinbock, and G.M. [10,13,27]. We actually make use of [27] to control how much mass of a closed H-orbit can be close to infinity, see Lemma 3.6.1.…”

Section: 7mentioning

confidence: 99%

“…More specifically, when applied with µ 1 = µ 2 , it yields an effective version of Lemma 2.5.1. 10 The generalization to two distinct measures µ 1 = µ 2 requires no effort, and is technically convenient for certain other applications (e.g., when studying two distinct closed S-orbits).…”

Section: 4mentioning

confidence: 99%

“…Dani has refined this in [10] to show that the corresponding conclusion remains valid, even without the constraint that the original point have low height, unless there is a rational subspace that is invariant under the unipotent one-parameter subgroup considered (i.e. a rational constraint prohibits that).…”

Section: A3 Intermediate Subgroupsmentioning

confidence: 99%

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Effective equidistribution for closed orbits of semisimple groups on homogeneous spaces

2009

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Section: 7mentioning

confidence: 99%

Section: 4mentioning

confidence: 99%

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Effective equidistribution for closed orbits of semisimple groups on homogeneous spaces

2009

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“…The following result, essentially due to Dani and Margulis [D1,DM2], is one of the most important results for studying unipotent flows on non-compact finite-volume homogeneous spaces. Remark.…”

Section: Flows On Finite-volume Homogeneous Spacesmentioning

confidence: 99%

On actions of epimorphic subgroups on homogeneous spaces

Shah

Weiss

2000

Ergod. Th. Dynam. Sys.

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Abstract. For an inclusion F < G < L of connected real algebraic groups such that F is epimorphic in G, we show that any closed F -invariant subset of L/ is G-invariant, where is a lattice in L. This is a topological analogue of a result due to S. Mozes, thatThis result is established by proving the following result. If in addition G is generated by unipotent elements, then there exists a ∈ F such that the following holds. Let U ⊂ F be the subgroup generated by all unipotent elements of F , x ∈ L/ , and λ and µ denote the Haar probability measures on the homogeneous spaces Ux and Gx, respectively (cf. Ratner's theorem). Then a n λ → µ weakly as n → ∞.We also give an algebraic characterization of algebraic subgroups F < SL n (R) for which all orbit closures on SL n (R)/SL n (Z) are finite-volume almost homogeneous, namely the smallest observable subgroup of SL n (R) containing F should have no nontrivial algebraic characters defined over R.

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“…This theorem is a refined version of Ratner's uniform distribution theorem [Rat4]; the proof uses Ratner's measure classification theorem (see [Rat1,Rat2,Rat3]), Dani's theorem on the behavior of unipotent orbits at infinity [Dan1,Dan2], and "linearization" techniques.…”

Section: Heref Is the Function On The Space Of Lattices Defined In (1mentioning

confidence: 99%

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Eskin¹,

Margulis²,

Mozes³

1995

Electron. Res. Announc. Amer. Math. Soc.

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Abstract. The Oppenheim conjecture, proved by Margulis in 1986, states that the set of values at integral points of an indefinite quadratic form in three or more variables is dense, provided the form is not proportional to a rational form. In this paper we study the distribution of values of such a form. We show that if the signature of the form is not (2, 1) or (2, 2), then the values are uniformly distributed on the real line, provided the form is not proportional to a rational form. In the cases where the signature is (2, 1) or (2, 2) we show that no such universal formula exists, and give asymptotic upper bounds which are in general best possible.Let Q be an indefinite nondegenerate quadratic form in n variables. Let L Q = Q(Z n ) denote the set of values of Q at integral points. The Oppenheim conjecture, proved by Margulis (cf. [Mar]) states that if n ≥ 3, and Q is not proportional to a form with rational coefficients, then L Q is dense. The Oppenheim conjecture enjoyed attention and many studies since it was conjectured in 1929 mostly using analytic number theory methods. In this paper 1 we study some finer questions related to the distribution the values of Q at integral points. 1.Let ν be a continuous positive function on the sphere {v ∈ R n | v = 1}, and let Ω = {v ∈ R n | v < ν(v/ v )}. We denote by T Ω the dilate of Ω by T . Define the following set: (a,b) when there is no confusion about the form Q. Also let V (a,b)

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On orbits of unipotent flows on homogeneous spaces

Cited by 67 publications

References 7 publications

Effective equidistribution for closed orbits of semisimple groups on homogeneous spaces

Effective equidistribution for closed orbits of semisimple groups on homogeneous spaces

On actions of epimorphic subgroups on homogeneous spaces

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