On orbits of unipotent flows on homogeneous spaces, II
On orbits of unipotent flows on homogeneous spaces, II
复制标题
关于均匀空间上的单能流轨道，II
DOI：
--
复制
发表时间：
1986
影响因子：
0.9
通讯作者：
S. Dani
中科院分区：
文献类型：
--
作者：
S. Dani
本文证明了：若（ut）是SL（n，n）的单参数幂单子群，则对任意ε > 0，存在SL（n，n）/SL（n，n）的紧子集K，使得对任意g ∈ SL（n，n），m（{t ∈ [0，T]| utg SL（n，n）∈ K}）>（1 - ε）T对于所有大T（m是勒贝格测度）或存在由有理方程定义的非平凡（g−1utg）-不变子空间。类似的结果推导出其他齐性空间上的幂幺流的轨道。如果G是连通半单李群，且G中的格为Γ，则存在G的紧子集D，使得对于G的任意闭连通幂幺子群U，且U不包含在G的任意真闭子群中，有G = DΓ U.将分解应用到丢番图逼近中。
Abstract We show that if (ut) is a one-parameter subgroup of SL (n, ℝ) consisting of unipotent matrices, then for any ε > 0 there exists a compact subset K of SL(n, ℝ)/SL(n, ℤ) such that the following holds: for any g ∈ SL(n, ℝ) either m({t ∈ [0, T] | utg SL (n, ℤ) ∈ K}) > (1 – ε)T for all large T (m being the Lebesgue measure) or there exists a non-trivial (g−1utg)-invariant subspace defined by rational equations. Similar results are deduced for orbits of unipotent flows on other homogeneous spaces. We also conclude that if G is a connected semisimple Lie group and Γ is a lattice in G then there exists a compact subset D of G such that for any closed connected unipotent subgroup U, which is not contained in any proper closed subgroup of G, we have G = DΓ U. The decomposition is applied to get results on Diophantine approximation.