齐性空间上流的对数律
Logarithm laws for flows on homogeneous spaces
摘要：
本文证明了有限体积的非紧局部对称空间上几乎所有测地线都以对数速度增长，这是D. Sullivan（1982）.更一般地说，在一定条件下的一个序列的子集$A_n$的齐次空间$G/Γ$（$G$半单李群，$Γ$非均匀格）和一个序列的元素$f_n$的$G$我们证明，几乎所有的点$x$的空间，一个有$f_n x\in A_n$无限多$n$。主要工具是$G/Γ$上平滑函数相关系数的指数衰减。除了上述对测地线流的应用外，作为推论，我们还得到了经典的Khinchin-Groshev定理在同时丢番图逼近中的一个新证明，并解决了M.斯克里加诺夫
英文摘要：
We prove that almost all geodesics on a noncompact locally symmetric space of finite volume grow with a logarithmic speed -- the higher rank generalization of a theorem of D. Sullivan (1982). More generally, under certain conditions on a sequence of subsets $A_n$ of a homogeneous space $G/Γ$ ($G$ a semisimple Lie group, $Γ$ a non-uniform lattice) and a sequence of elements $f_n$ of $G$ we prove that for almost all points $x$ of the space, one has $f_n x\in A_n$ for infinitely many $n$. The main tool is exponential decay of correlation coefficients of smooth functions on $G/Γ$. Besides the aforementioned application to geodesic flows, as a corollary we obtain a new proof of the classical Khinchin-Groshev theorem in simultaneous Diophantine approximation, and settle a related conjecture recently made by M. Skriganov.
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