Invariant and stationary measures for the $\mathrm{SL}(2,ℝ)$ action on Moduli space

Eskin, Alex; Mirzakhani, Maryam

Publications Mathématiques de l'IHÉS, Tome 127 (2018), pp. 95-324.

We prove some ergodic-theoretic rigidity properties of the action of $\mathrm{SL}(2,ℝ)$ on moduli space. In particular, we show that any ergodic measure invariant under the action of the upper triangular subgroup of $\mathrm{SL}(2,ℝ)$ 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.

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