## Uniform distribution of orbits of lattices on spaces of frames.(English)Zbl 1067.37004

Let $$G =\text{SL}(n,\mathbb R)$$ and $$V_{n,l}$$ be the space of $$l$$-frames in $$\mathbb R^n$$, $$1\leq l\leq n$$. The group $$G$$ acts on this space as follows: $$g(\nu_1,\dots,\nu_l)=(g\nu_1,\dots,g\nu_l)$$, $$g\in G$$ and $$(\nu_1,\dots,\nu_n)\in V_{n,l}$$. Let $$\Gamma$$ be a lattice in $$G$$ such that $$\Gamma\setminus G$$ has finite volume. The main result of this paper concerns the distribution of $$\Gamma$$-orbits in $$V_{n,l}$$. When $$l = n$$, every orbit of $$\Gamma$$ is discrete. Let $$l< n$$, $$F = \text{SL}(n, Z)$$ and $$\nu = (\nu_1,\dots,\nu_n)$$ be an $$l$$-frame in $$\mathbb R^n$$. S. G. Dani and S. Raghavan [Isr. J. Math. 36, 300–320 (1980; Zbl 0457.28008)] proved that the orbit $$\Gamma.\nu$$ is dense in $$V_{n,l}$$ if and only if the space spanned by $$(\nu_i, i =1,\dots,l)$$ contains no nonzero rational vectors. W. A. Veech [Am. J. Math. 99, 827–859 (1977; Zbl 0365.28012)] has shown that if $$\Gamma$$ is a cocompact lattice in $$G$$, then every orbit in $$V_{n,l}$$ $$l < n$$ is dense. Here, it is shown that dense $$\Gamma$$-orbits are uniformly distributed with respect to an explicitly described measure on $$V_{n,l}$$. Dani and Raghvan also considered orbits of frames under $$\text{Sp}(n,\mathbb Z)$$. They proved that for an isotropic frame $$\nu = (\nu_1,\dots,\nu_n)$$ in $$\mathbb R^{2n}$$, $$\Gamma.\nu$$ is dense in the space of isotropic $$n$$-frames if and only if the space spanned by $$(\nu_i$$, $$i =1,\dots,n)$$ contains no nonzero rational vectors. Here, this result is improved by showing that the dense orbits of $$\Gamma$$ are uniformly distributed. The method of the proof uses the Iwasawa decomposition for $$\text{Sp}(n,\mathbb R)$$ and uniform dist of large unipotent subgroups due to Shah. A proof for the results of Dani, Raghvan and Veech based on topological rigidity of unipotent flows is also given.

### MSC:

 37A15 General groups of measure-preserving transformations and dynamical systems 22E40 Discrete subgroups of Lie groups 37A17 Homogeneous flows 20H05 Unimodular groups, congruence subgroups (group-theoretic aspects)

### Citations:

Zbl 0365.28012; Zbl 0457.28008
Full Text:

### References:

 [1] S. G. Dani and G. A. Margulis, “Limit distributions of orbits of unipotent flows and values of quadratic forms” in I. M. Gelfand Seminar , Adv. Soviet Math. 16 , Part 1, Amer. Math. Soc., Providence, 1993, 91-137. · Zbl 0814.22003 [2] S. G. Dani and S. Raghavan, Orbits of Euclidean frames under discrete linear groups , Israel J. Math. 36 (1980), 300-320., 1980. · Zbl 0457.28008 [3] W. Duke, Z. Rudnick, and P. Sarnak, Density of integer points on affine homogeneous varieties , Duke Math. J. 71 (1993), 143-179. · Zbl 0798.11024 [4] A. Gorodnik, Lattice action on the boundary of $$\SL(n,\mathbb{R})$$ , Ergodic Theory Dynam. Systems 23 (2003), 1817-1837. · Zbl 1050.22015 [5] S. Helgason, Differential Geometry and Symmetric Spaces , Pure Appl. Math. 12 , Academic Press, New York, 1962. · Zbl 0111.18101 [6] D. Kleinbock, N. Shah, and A. Starkov, “Dynamics of subgroup actions on homogeneous spaces of Lie groups and applications to number theory” in Handbook of Dynamical Systems, Vol. 1A , North-Holland, Amsterdam, 2002, 813-930. · Zbl 1050.22026 [7] F. Ledrappier, Distribution des orbites des réseaux sur le plan réel , C. R. Acad. Sci. Paris Sér. I Math. 329 (1999), 61-64. · Zbl 0928.22012 [8] A. Nogueira, Orbit distribution on $$\mathbb{R}^2$$ under the natural action of $$\SL(2,\mathbb{Z})$$ , Indag. Math. (N.S.) 13 (2002), 103-124. · Zbl 1016.37003 [9] M. S. Raghunathan, Discrete Subgroups of Lie Groups , Ergeb. Math. Grenzgeb. (2) 68 , Springer, New York, 1972. · Zbl 0254.22005 [10] M. Ratner, On Raghunathan’s measure conjecture , Ann. of Math. (2) 134 (1991), 545-607. JSTOR: · Zbl 0763.28012 [11] -. -. -. -., Raghunathan’s topological conjecture and distributions of unipotent flows , Duke Math. J. 63 (1991), 235-280. · Zbl 0733.22007 [12] N. A. Shah, Uniformly distributed orbits of certain flows on homogeneous spaces , Math. Ann. 289 (1991), 315-334. · Zbl 0702.22014 [13] -. -. -. -., Limit distributions of polynomial trajectories on homogeneous spaces , Duke Math. J. 75 (1994), 711-732. · Zbl 0818.22005 [14] -. -. -. -., Limit distributions of expanding translates of certain orbits on homogeneous spaces , Proc. Indian Acad. Sci. Math. Sci. 106 (1996), 105-125. · Zbl 0864.22004 [15] -. -. -. -., “Counting integral matrices with a given characteristic polynomial” in Ergodic Theory and Harmonic Analysis (Mumbai, 1999) , Sankhyā Ser. A 62 (2002), 386-412. · Zbl 1002.22007 [16] N. A. Shah and B. Weiss, On actions of epimorphic subgroups on homogeneous spaces , Ergodic Theory Dynam. Systems 20 (2002), 567-592. · Zbl 0949.22012 [17] A. N. Starkov, Dynamical Systems on Homogeneous Spaces , Transl. Math. Monogr. 190 , Amer. Math. Soc., Providence, 2000. · Zbl 1143.37300 [18] A. Terras, Harmonic Analysis on Symmetric Spaces and Applications, II , Springer, Berlin, 1988. · Zbl 0668.10033 [19] W. A. Veech, Unique ergodicity of horospherical flows , Amer. J. Math. 99 (1977), 827-859. JSTOR: · Zbl 0365.28012 [20] D. V. Widder, The Laplace Transform , Princeton Math. Ser. 6 , Princeton Univ. Press, Princeton, 1941. · Zbl 0429.46016
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