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

Eskin, Alex; Mirzakhani, Maryam

doi:10.1007/s10240-018-0099-2

Invariant and stationary measures for the

SL (2, ℝ)

action on Moduli space

Eskin, Alex ; Mirzakhani, Maryam

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

Résumé

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

DOI : 10.1007/s10240-018-0099-2

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     title = {Invariant and stationary measures for the $\mathrm{SL} (2 , \mathbb{R})$ action on {Moduli} space},
     journal = {Publications Math\'ematiques de l'IH\'ES},
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Eskin, Alex; Mirzakhani, Maryam. Invariant and stationary measures for the $\mathrm{SL} (2 , \mathbb{R})$ action on Moduli space. Publications Mathématiques de l'IHÉS, Tome 127 (2018), pp. 95-324. doi : 10.1007/s10240-018-0099-2. http://www.numdam.org/articles/10.1007/s10240-018-0099-2/

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