[Paper Review] SL(2,R)-invariant probability measures on the moduli spaces of translation surfaces are regular

Artur Avila, Carlos Matheus+1 more|arXiv (Cornell University)|Feb 17, 2013

Mathematical Dynamics and Fractals13 references4 citations

TL;DR

This paper proves that any $SL(2,\mathbb{R})$-invariant probability measure on the moduli space of normalized translation surfaces of genus $g$ is regular, by showing that the set of surfaces with two non-parallel saddle-connections of length $\leq \rho$ has measure $o(\rho^2)$ as $\rho \to 0$. The result resolves a key technical assumption in the Eskin-Kontsevich-Zorich formula for Lyapunov exponents.

ABSTRACT

In the moduli space $H_g$ of normalized translation surfaces of genus $g$, consider, for a small parameter $ρ>0$, those translation surfaces which have two non-parallel saddle-connections of length $\leq ρ$. We prove that this subset of $H_g$ has measure $o(ρ^2)$ w.r.t. any probability measure on $H_g$ which is invariant under the natural action of $SL(2,R)$. This implies that any such probability measure is regular, a property which is important in relation with the recent fundamental work of Eskin-Kontsevich-Zorich on the Lyapunov exponents of the KZ-cocycle.

Motivation & Objective

• To establish the regularity of $SL(2,\mathbb{R})$-invariant probability measures on the moduli space $\mathcal{H}_g$ of normalized translation surfaces of genus $g \geq 1$.
• To resolve a technical gap in the Eskin-Kontsevich-Zorich formula for sums of Lyapunov exponents, which requires the measure to be regular.
• To show that the set of translation surfaces with two non-parallel saddle-connections of length $\leq \rho$ has measure $o(\rho^2)$ under any such invariant measure.
• To provide a geometric and dynamical justification for the integration by parts argument used in the Eskin-Kontsevich-Zorich formula.

Proposed method

• Use the $SL(2,\mathbb{R})$ action to analyze the dynamics of saddle-connections via the decomposition $g_t R_\theta n_u$.
• Apply conditional measure techniques to study the behavior of measures along orbits of the diagonal subgroup $g_t = \mathrm{diag}(e^t, e^{-t})$.
• Estimate the norm of vectors under $SL(2,\mathbb{R})$ transformations using Euclidean norms and trigonometric identities.
• Construct a reference measure $m_0$ on a fundamental domain $X_0^*$ and relate it to the full invariant measure $m$ via scaling and integration.
• Use angular separation and minimal angle estimates between saddle-connections to control the contribution of singular configurations.
• Apply integration by parts and curvature-based arguments to relate geometric quantities to Siegel-Veech constants, relying on the regularity condition.

Experimental results

Research questions

• RQ1 Does every $SL(2,\mathbb{R})$-invariant probability measure on $\mathcal{H}_g$ satisfy the regularity condition required for the Eskin-Kontsevich-Zorich formula?
• RQ2 What is the asymptotic measure of the set of translation surfaces with two non-parallel saddle-connections of length $\leq \rho$?
• RQ3 How does the dynamics of the $SL(2,\mathbb{R})$ action control the geometry of short saddle-connections?
• RQ4 Can the integration by parts argument in the Eskin-Kontsevich-Zorich formula be justified without additional assumptions?
• RQ5 What is the role of angular separation between short saddle-connections in the regularity of invariant measures?

Key findings

• The set of translation surfaces with two non-parallel saddle-connections of length $\leq \rho$ has measure $o(\rho^2)$ with respect to any $SL(2,\mathbb{R})$-invariant probability measure on $\mathcal{H}_g$.
• This implies that any such invariant measure is regular, as defined in the Eskin-Kontsevich-Zorich work.
• The regularity condition is satisfied because the measure of surfaces with short, non-parallel saddle-connections decays faster than $\rho^2$.
• The proof relies on controlling the norm of vectors under $g_t R_\theta$ actions and using angular separation to exclude problematic configurations.
• The result confirms that the integration by parts argument in the Eskin-Kontsevich-Zorich formula is valid under $SL(2,\mathbb{R})$-invariance alone.
• The key estimate is $m(Y(T,\frac{\pi}{2},X_0^*)) < \frac{1}{4}\eta \rho^2$ and similar bounds, which together imply the $o(\rho^2)$ decay.