We prove some ergodic-theoretic rigidity properties of the action of on moduli space. In particular, we show that any ergodic measure invariant under the action of the upper triangular subgroup of is supported on an invariant affine submanifold.

The main theorems are inspired by the results of several authors on unipotent flows on homogeneous spaces, and in particular by Ratner’s seminal work.

中文翻译：

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Moduli 空间上作用的不变和平稳测度

我们证明了模空间作用的一些遍历理论刚性性质。特别是，我们证明了在 的上三角子群作用下的任何遍历测度不变都在不变仿射子流形上得到支持。

主要定理的灵感来自于几位作者关于齐次空间上的单能流的结果，特别是拉特纳的开创性工作。