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

Abstract

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.

arXiv comment214 pages, 8 figures. Final version, to appear in Publications l'IHES

References in corpus (3)

• Ratner's Theorems on Unipotent Flows
• Lattice Point Asymptotics and Volume Growth on Teichmuller space
• An example of a Teichmuller disk in genus 4 with degenerate Kontsevich-Zorich spectrum

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• Everything is illuminated
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• The moduli space of multi-scale differentials
• Full rank affine invariant submanifolds
• Marked points on translation surfaces
• Invariant measures and measurable projective factors for actions of higher-rank lattices on manifolds
• Masur-Veech volumes and intersection theory: the principal strata of quadratic differentials
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• GL(2,R)-Invariant Measures in Marked Strata: Generic Marked Points, Earle-Kra for Strata, and Illumination
• Complete periodicity of Prym eigenforms
• Non-Veech surfaces in H^hyp(4) are generic
• Affine geometry of strata of differentials
• Eigenvalues of Curvature, Lyapunov exponents and Harder-Narasimhan filtrations
• Recent progress in the Zimmer program
• Quasimodularity and large genus limits of Siegel-Veech constants
• Degenerations of Abelian Differentials
• Equations of linear subvarieties of strata of differentials
• Square-tiled surfaces of fixed combinatorial type: equidistribution, counting, volumes of the ambient strata
• Teichmüller dynamics in the eyes of an algebraic geometer
• On Hoelder-continuity of Oseledets subspaces
• The boundary of linear subvarieties
• A short introduction to translation surfaces, Veech surfaces, and Teichmueller dynamics
• Teichmueller curves in genus three and just likely intersections in $G_m^{n}x{G}_a^n$
• Zero Lyapunov exponents of the Hodge bundle
• On the effective cone of ${\overline{\mathcal{M}}}_{g,n}$
• Loci in strata of meromorphic differentials with fully degenerate Lyapunov spectrum
• Counting problem on wind-tree models
• Reconstructing orbit closures from their boundaries
• Shear-shape cocycles for measured laminations and ergodic theory of the earthquake flow
• Measure bound for translation surfaces with short saddle connections
• A non-varying phenomenon with an application to the wind-tree model
• Hodge and Teichmüller
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• Measure rigidity for random dynamics on surfaces with positive entropy
• Invariant Subvarieties of Minimal Homological Dimension, Zero Lyapunov Exponents, and Monodromy
• The Magic Wand Theorem of A. Eskin and M. Mirzakhani
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• Uniqueness of $u$-Gibbs measures for hyperbolic skew products on ${\mathbb{T}}^4$
• Effective Unique Ergodicity and Weak Mixing of Translation Flows
• Billiards in right triangles and orbit closures in genus zero strata
• Spin and hyperelliptic structures of log twisted differentials
• Effective divisors in ${\overline{\mathcal{M}}}_{g,n}$ from abelian differentials
• Quantitative error term in the counting problem on Veech wind-tree models
• Non-existence and finiteness results for Teichmueller curves in Prym loci
• A coding-free simplicity criterion for the Lyapunov exponents of Teichmueller curves
• Tautological ring of strata of differentials
• Convergence of Siegel-Veech constants
• Quelques contributions à la théorie de l'action de SL(2,R) sur les espaces de modules de surfaces plates
• On Convergence of Random Walks on Moduli Space
• Non-affine horocycle orbit closures on strata of translation surfaces: new examples
• A Central Limit Theorem for the Kontsevich-Zorich Cocycle
• Spherical averages in the space of marked lattices