The dynamics of one-parameter diagonal group actions on finite volume homogeneous spaces has a partially hyperbolic feature. In this paper we extend the Livšic-type result to these possibly noncompact and nonaccessible systems. We also prove a central limit theorem for the Birkhoff averages of points on a horospherical orbit. The Livšic-type result allows us to show that the variance of the central limit theorem is nonzero provided that the test function has nonzero mean with respect to an invariant probability measure.

中文翻译：

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齐次空间上的中心极限定理和同调方程

有限体积齐次空间上的一参数对角线群动作的动力学具有部分双曲特征。在本文中，我们将Livšic类型的结果扩展到这些可能非紧凑且不可访问的系统。我们还证明了球面轨道上伯克霍夫（Birkhoff）平均点的中心极限定理。Livšic类型的结果使我们能够证明，如果检验函数相对于不变概率测度具有非零均值，则中心极限定理的方差为非零。