Asymptotic behaviour of trajectories of unipotent flows on homogeneous spaces
Asymptotic behaviour of trajectories of unipotent flows on homogeneous spaces
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齐次空间上单能流轨迹的渐近行为
DOI：
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发表时间：
1991
期刊：
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通讯作者：
G. Margulis
中科院分区：
文献类型：
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作者：
S. Dani;G. Margulis
证明了：若G是定义在Q上的半单代数群，且G：=GR中的算术格是关于Q-结构的，则存在G/Γ的紧子集C，使得对G的任意幂单参数子群{ut}和任意g ∈G，g Γ的{ut}-轨迹在时间区间[0，T]内在C中所花费的时间渐近于T，除非{g−1utg}包含在G的Q-抛物子群中.这也证明了一些定量的版本。这些结果加强了文献[5]中对SL（n，Z），n≥2的类似断言，并且也验证了文献[7]中对SL（3，R）中的格所引入的一个技术条件，该条件在文献[8]中用于证明Raghunathan关于一类幂单流的猜想。
We show that ifG is a semisimple algebraic group defined overQ and Γ is an arithmetic lattice inG:=GR with respect to theQ-structure, then there exists a compact subsetC ofG/Γ such that, for any unipotent one-parameter subgroup {ut} ofG and anyg∈G, the time spent inC by the {ut}-trajectory ofgΓ, during the time interval [0,T], is asymptotic toT, unless {g−1utg} is contained in aQ-parabolic subgroup ofG. Some quantitative versions of this are also proved. The results strengthen similar assertions forSL(n,Z),n≥2, proved earlier in [5] and also enable verification of a technical condition introduced in [7] for lattices inSL(3,R), which was used in our proof of Raghunathan’s conjecture for a class of unipotent flows, in [8].