The distribution of periodic torus orbits on homogeneous spaces
The distribution of periodic torus orbits on homogeneous spaces
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均匀空间上周期环面轨道的分布
DOI：
10.1215/00127094-2009-023
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发表时间：
2006
影响因子：
2.5
通讯作者：
Akshay Venkatesh
中科院分区：
文献类型：
--
作者：
M. Einsiedler;E. Lindenstrauss;P. Michel;Akshay Venkatesh
我们证明了齐次空间上某些周期环面轨道族的均匀分布，重点讨论了对角环面作用于PGLn(R)商的情况。在给每个周期轨道附加一个积分不变量（区别式）之后，我们的结果有如下的味道：关于这些轨道分布的某些标准猜想支持最多O(O)个区别≤的轨道的例外集。该证明依赖于周期轨道的良好分离性以及环面作用的测量刚度。我们也给出了这个作用的周期轨道序列的例子，即使在更高的秩上，它们也不能成为等分布的。在三次及以上次的全实数域上，利用我们的结果锐化了闵可夫斯基在理想类上的一个定理。
We prove results towards the equidistribution of certain families of periodic torus orbits on homogeneous spaces, with particular focus on the case of the diagonal torus acting on quotients of PGLn(R). After attaching to each periodic orbit an integral invariant (the discriminant) our results have the following flavour: certain standard conjectures about the distribution of such orbits hold up to exceptional sets of at most O(� ǫ ) orbits of discriminant ≤ �. The proof relies on the well-separatedness of periodic orbits together with measure rigidity for torus actions. We also give examples of sequences of periodic orbits of this action that fail to become equidistributed, even in higher rank. We give an application of our results to sharpen a theorem of Minkowski on ideal classes in totally real number fields of cubic and higher degrees.