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.

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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     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},
     year = {2018},
     publisher = {Springer Berlin Heidelberg},
     address = {Berlin/Heidelberg},
     volume = {127},
     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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