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.
205 Citations
Invariant and stationary measures for the SL(2,R) action on Moduli space
- Mathematics
- 2013
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…
Invariant Subvarieties of Minimal Homological Dimension, Zero Lyapunov Exponents, and Monodromy
- Mathematics
- 2021
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 and…
The WYSIWYG compactification
- MathematicsJournal of the London Mathematical Society
- 2020
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 of…
Reconstructing orbit closures from their boundaries
- Mathematics
- 2020
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 simultaneously…
GL+(2, ℝ)–orbits in Prym eigenform loci
- Mathematics
- 2013
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 the…
Classification of higher rank orbit closures in H^{odd}(4)
- Mathematics
- 2013
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 closed…
Compactification of strata of Abelian differentials
- MathematicsDuke Mathematical Journal
- 2018
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 with…
Counting special Lagrangian classes and semistable Mukai vectors for K3 surfaces
- Mathematics
- 2021
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 an…
Full-rank affine invariant submanifolds
- MathematicsDuke Mathematical Journal
- 2018
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 extra…
The boundary of an affine invariant submanifold
- Mathematics
- 2015
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 Riemann…
References
SHOWING 1-10 OF 104 REFERENCES
Invariant and stationary measures for the SL(2,R) action on Moduli space
- Mathematics
- 2013
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…
Effective equidistribution for closed orbits of semisimple groups on homogeneous spaces
- Mathematics
- 2007
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 should…
Symplectic and Isometric SL(2,R) invariant subbundles of the Hodge bundle
- Mathematics
- 2012
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 maximal…
An analytic construction of the Deligne-Mumford compactification of the moduli space of curves
- Mathematics
- 2013
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.…
Lyapunov spectrum of invariant subbundles of the Hodge bundle
- MathematicsErgodic Theory and Dynamical Systems
- 2012
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 moduli…
Every flat surface is Birkhoff and Oseledets generic in almost every direction
- Mathematics
- 2013
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 large…
Flows on homogeneous spaces and Diophantine approximation on manifolds
- Mathematics
- 1998
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 homogeneous…
On orbits of unipotent flows on homogeneous spaces
- MathematicsErgodic Theory and Dynamical Systems
- 1984
Abstract Let G be a connected Lie group and let Γ be a lattice in G (not necessarily co-compact). We show that if (ut) is a unipotent one-parameter subgroup of G then every ergodic invariant (locally…
Geometry of the Weil-Petersson completion of Teichm\
- Mathematics
- 2003
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 and…
On the space of ergodic invariant measures of unipotent flows
- MathematicsErgodic Theory and Dynamical Systems
- 1995
Abstract Let G be a Lie group and Γ be a discrete subgroup. We show that if {μn} is a convergent sequence of probability measures on G/Γ which are invariant and ergodic under actions of unipotent…