Isolation, equidistribution, and orbit closures for the SL(2,R) action on Moduli space

@article{Eskin2013IsolationEA,
  title={Isolation, equidistribution, and orbit closures for the SL(2,R) action on Moduli space},
  author={A. V. Eskin and Maryam Mirzakhani and A. Mohammadi},
  journal={arXiv: Dynamical Systems},
  year={2013}
}
We prove results about orbit closures and equidistribution for the SL(2,R) action on the moduli space of compact Riemann surfaces, which are analogous to the theory of unipotent flows. The proofs of the main theorems rely on the measure classification theorem of [EMi2] and a certain isolation property of closed SL(2,R) invariant manifolds developed in this paper. 
Invariant and stationary measures for the SL(2,R) action on Moduli space
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 triangularExpand
The WYSIWYG compactification
We show that the partial compactification of a stratum of Abelian differentials previously considered by Mirzakhani and Wright is not an algebraic variety. Despite this, we use a combination ofExpand
Invariant Subvarieties of Minimal Homological Dimension, Zero Lyapunov Exponents, and Monodromy
We classify the GL(2,R)-invariant subvarieties M in strata of Abelian differentials for which any two M-parallel cylinders have homologous core curves. This answers a question of Mirzakhani andExpand
Reconstructing orbit closures from their boundaries
We introduce and study diamonds of GL(2,R)-invariant subvarieties of Abelian and quadratic differentials, which allow us to recover information on an invariant subvariety by simultaneouslyExpand
GL+(2, ℝ)–orbits in Prym eigenform loci
This paper is devoted to the classification of GL^+(2,R)-orbit closures of surfaces in the intersection of the Prym eigenform locus with various strata of quadratic differentials. We show that theExpand
Classification of higher rank orbit closures in H^{odd}(4)
The moduli space of genus 3 translation surfaces with a single zero has two connected components. We show that in the odd connected component H^{odd}(4) the only GL^+(2,R) orbit closures are closedExpand
Compactification of strata of Abelian differentials
We describe the closure of the strata of abelian differentials with prescribed type of zeros and poles, in the projectivized Hodge bundle over the Deligne-Mumford moduli space of stable curves withExpand
Counting special Lagrangian classes and semistable Mukai vectors for K3 surfaces
Motivated by the study of growth rate of the number of geodesics in flat surfaces with bounded lengths, we study generalizations of such problems for K3 surfaces. In one generalization, we give anExpand
Full rank affine invariant submanifolds
We show that every GL(2, R) orbit closure of translation surfaces is either a connected component of a stratum, the hyperelliptic locus, or consists entirely of surfaces whose Jacobians have extraExpand
The boundary of an affine invariant submanifold
We study the boundary of an affine invariant submanifold of a stratum of translation surfaces in a partial compactification consisting of all finite area Abelian differentials over nodal RiemannExpand
...
1
2
3
4
5
...

References

SHOWING 1-10 OF 104 REFERENCES
Invariant and stationary measures for the SL(2,R) action on Moduli space
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 triangularExpand
Effective equidistribution for closed orbits of semisimple groups on homogeneous spaces
We prove effective equidistribution, with polynomial rate, for large closed orbits of semisimple groups on homogeneous spaces, under certain technical restrictions (notably, the acting group shouldExpand
Symplectic and Isometric SL(2,R) invariant subbundles of the Hodge bundle
Suppose N is an affine SL(2,R)-invariant submanfold of the moduli space of pairs (M,w) where M is a curve, and w is a holomorphic 1-form on M. We show that the Forni bundle of N (i.e. the maximalExpand
An analytic construction of the Deligne-Mumford compactification of the moduli space of curves
In 1969, P. Deligne and D. Mumford compactified the moduli space of curves. Their compactification is a projective algebraic variety, and as such, it has an underlying analytic structure.Expand
Lyapunov spectrum of invariant subbundles of the Hodge bundle
Abstract We study the Lyapunov spectrum of the Kontsevich–Zorich cocycle on SL(2,ℝ)-invariant subbundles of the Hodge bundle over the support of SL(2,ℝ)-invariant probability measures on the moduliExpand
Every flat surface is Birkhoff and Oseledets generic in almost every direction
We prove that the Birkhoff pointwise ergodic theorem and the Oseledets multiplicative ergodic theorem hold for every flat surface in almost every direction. The proofs rely on the strong law of largeExpand
Flows on homogeneous spaces and Diophantine approximation on manifolds
We present a new approach to metric Diophantine approximation on manifolds based on the correspondence between approximation properties of numbers and orbit properties of certain flows on homogeneousExpand
Geometry of the Weil-Petersson completion of Teichm\
We present a view of the current understanding of the geometry of Weil-Petersson (WP) geodesics on the completion of the Teichm\"uller space. We sketch a collection of results by other authors andExpand
The field of definition of affine invariant submanifolds of the moduli space of abelian differentials
The field of definition of an affine invariant submanifold M is the smallest subfield of R such that M can be defined in local period coordinates by linear equations with coefficients in this field.Expand
On the space of ergodic invariant measures of unipotent flows
Let G be a Lie group and Γ be a discrete subgroup. We show that if {μ n } is a convergent sequence of probability measures on G/Γ which are invariant and ergodic under actions of unipotentExpand
...
1
2
3
4
5
...