Invariant and stationary measures for the SL(2,R) action on Moduli space (1302.3320v6)

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.

• The paper establishes that any ergodic measure invariant under the SL(2,R) action is supported on an affine invariant submanifold, paralleling unipotent flow results.
• It employs entropy methods and explicit semi-Markov constructions to analyze dynamical systems, advancing techniques in measure classification.
• The study offers practical insights by deriving asymptotic formulas for counting periodic billiard trajectories through Siegel-Veech constants.

Overview of the Paper on Invariant and Stationary Measures for the $SL(2,\mathbb{R})$ Action on Moduli Space

This paper, authored by Alex Eskin and Maryam Mirzakhani, focuses on the ergodic-theoretic properties of $SL(2,\mathbb{R})$ actions on moduli spaces of Abelian differentials. Central to the work is the understanding of the behavior of invariant and stationary measures under the $SL(2,\mathbb{R})$ group action on these moduli spaces, addressing a fundamental problem in the area of ergodic theory and homogeneous dynamics.

The paper draws inspiration from Ratner’s theorems on unipotent flows, emphasizing the analogous rigidity phenomena in the context of moduli spaces. The main contributions include results on the classification of measures invariant under subgroups of $SL(2,\mathbb{R})$, particularly highlighting the role of affine invariant submanifolds and extending the framework of unipotent flows to this new setting.

Key Results

1. Invariant Measures on Moduli Spaces:
   • The paper demonstrates that any ergodic measure invariant under the action of the upper triangular subgroup of $SL(2,\mathbb{R})$ is supported on a subset that is an affine invariant submanifold.
   • This result parallels well-known facts about unipotent flows on homogeneous spaces and applies them in a novel setting, moduli spaces.
2. Structure of Affine Invariant Submanifolds:
   • An affine invariant submanifold is defined in terms of its characterization through the Lyapunov exponents of the $SL(2,\mathbb{R})$ action, establishing foundational understanding for future classification efforts.
   • The submanifolds are shown to be defined over $\mathbb{Q}$, leading to conjectures about their classification in higher-genus surfaces.
3. Equidistribution and Counting Problems:
   • The work addresses applications to counting problems in rational billiards by establishing weak asymptotic formulas for the number of periodic trajectories.
   • Siegel-Veech constants emerge in this context, linking the paper to classical problems in the geometry of numbers.
4. Methods and Techniques:
   • The authors utilize entropy methods coupled with explicit constructions of semi-Markov sections to analyze dynamical systems, extending recent techniques developed for partially hyperbolic systems.
   • A novel Jordan Canonical Form is employed to manage the complexity of the cocycle dynamics, offering a more accessible approach to understanding the invariant and stationary measures.

Theoretical and Practical Implications

The theoretical implications of this work are widespread, influencing the paper of ergodic theory, geometric topology, and the geometry of moduli spaces. By building a bridge between dynamics on homogeneous spaces and moduli spaces, the paper offers a fresh perspective on classical problems, presenting robust methods that have the potential to be generalized to other groups beyond $SL(2,\mathbb{R})$.

Practically, the results lay groundwork for advances in understanding the qualitative behavior of geodesic flows on moduli spaces and have applications in number theory, particularly through connections to the distribution of rational points on manifolds that model billiard paths in rational polygons.

Future Directions

The paper opens up several avenues for future research:

• Classification of Affine Invariant Submanifolds: Comprehensive classification remains open, especially for strata in genus three or higher.
• Refinement of Entropy Methods: Developing sharper tools to handle partially hyperbolic systems beyond current limitations.
• Exploration of Non-linear Actions: Extending the methods to other group actions and investigating the interplay with nontrivial morphologies of the underlying moduli spaces.

Eskin and Mirzakhani’s work illuminates the intricate structures sitting between dynamics and geometry, pushing boundaries towards a unified view of moduli spaces through ergodic theory, laying a foundational stone for the future of mathematical research in this vibrant area.