Geodesic
Invariant and stationary measures for the SL ( 2 , ℝ ) action on Moduli space
Publications Mathématiques de l'IHÉS, Tome 127 (2018), pp. 95-324

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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.

@article{PMIHES_2018__127__95_0,
     author = {Eskin, Alex and Mirzakhani, Maryam},
     title = {Invariant and stationary measures for the $\mathrm{SL} (2 , \mathbb{R})$ action on {Moduli} space},
     journal = {Publications Math\'ematiques de l'IH\'ES},
     pages = {95--324},
     publisher = {Springer Berlin Heidelberg},
     address = {Berlin/Heidelberg},
     volume = {127},
     year = {2018},
     doi = {10.1007/s10240-018-0099-2},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1007/s10240-018-0099-2/}
}
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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

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