Invariant and stationary measures for the SL(2,R) action on Moduli space

By A. Eskin, Maryam Mirzakhani. 2013

258 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/4c86aee3905f494f661467765dc4be22ff9dd05a

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In Applied Mathematics

1. Dawei Chen
2. Maryam Mirzakhani
3. Don Zagier
4. Martin Möller
5. Martin Moeller
6. A. Eskin
7. A. Sauvaget
8. A. Mohammadi

In Mathematical Physics

1. Dawei Chen
2. Martin Möller
3. Don Zagier
4. Martin Moeller
5. Maryam Mirzakhani
6. A. Sauvaget
7. A. Eskin
8. A. Wright

Citations in the corpus

• Masur–Veech volumes and intersection theory on moduli spaces of Abelian differentials (2019, background): “(88) There is a natural action of GL2(R) on the moduli space of flat surfaces Mg and the orbit closures are nice submanifolds, in fact linear in period coordinates by the fundamental work of Eskin, Mirzakhani and Mohammadi [17,18].”
• Isolation, equidistribution, and orbit closures for the SL(2,R) action on Moduli space (2013, result): “The following theorem is the main result of [EM13]: Theorem 1.3.”
• Quasimodularity and large genus limits of Siegel-Veech constants (2016, background): “Significant progress has been made recently by Eskin-Mirzakhani-Mohammadi ([16], [17]) by showing that orbit closures have a nice geometric structure, i.e. they are linear submanifolds of ΩMg.”