Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A Central Limit Theorem for the Kontsevich-Zorich Cocycle

Hamid Al-Saqban, Giovanni Forni

TL;DR

This work proves a Central Limit Theorem for exterior powers of the Kontsevich-Zorich cocycle over SL(2,R) orbit closures, applicable to both random (leafwise hyperbolic Brownian motion) and deterministic (Teichmüller geodesic flow) settings under a Lyapunov-spectrum gap assumption. The authors reduce the problem to solving a leafwise Poisson equation, construct a martingale from gradient differences, and apply a martingale CLT to obtain Gaussian limits with explicitly characterized variances. They establish positivity of the asymptotic variance in the random case (and for the top exponent of the deterministic cocycle under simplicity) via a potential-theoretic and geometric analysis, including harmonic measures and Ancona-type estimates. The results extend to future-Oseledets-generic sections, yielding CLTs for generic choices of vectors in the Hodge bundle and reinforcing the probabilistic understanding of Teichmüller dynamics in non-uniformly hyperbolic settings.

Abstract

We show that a central limit theorem holds for exterior powers of the Kontsevich-Zorich (KZ) cocycle. In particular, we show that, under the hypothesis that the top Lyapunov exponent on the exterior power is simple, a central limit theorem holds for the lift of the (leafwise) hyperbolic Brownian motion to any strongly irreducible, symplectic, $\text{SL}(2,\mathbb{R})$-invariant subbundle, that is moreover symplectic-orthogonal to the so-called tautological subbundle. We then show that this implies that a central limit theorem holds for the lift of the Teichmüller geodesic flow to the same bundle. For the random cocycle over the hyperbolic Brownian motion, we prove under the same hypotheses that the variance of the top exponent is strictly positive. For the deterministic cocycle over the Teichmüller geodesic flow we prove that the variance is strictly positive only for the top exponent of the first exterior power (the KZ cocycle itself) under the hypothesis that its Lyapunov spectrum is simple.

A Central Limit Theorem for the Kontsevich-Zorich Cocycle

TL;DR

This work proves a Central Limit Theorem for exterior powers of the Kontsevich-Zorich cocycle over SL(2,R) orbit closures, applicable to both random (leafwise hyperbolic Brownian motion) and deterministic (Teichmüller geodesic flow) settings under a Lyapunov-spectrum gap assumption. The authors reduce the problem to solving a leafwise Poisson equation, construct a martingale from gradient differences, and apply a martingale CLT to obtain Gaussian limits with explicitly characterized variances. They establish positivity of the asymptotic variance in the random case (and for the top exponent of the deterministic cocycle under simplicity) via a potential-theoretic and geometric analysis, including harmonic measures and Ancona-type estimates. The results extend to future-Oseledets-generic sections, yielding CLTs for generic choices of vectors in the Hodge bundle and reinforcing the probabilistic understanding of Teichmüller dynamics in non-uniformly hyperbolic settings.

Abstract

We show that a central limit theorem holds for exterior powers of the Kontsevich-Zorich (KZ) cocycle. In particular, we show that, under the hypothesis that the top Lyapunov exponent on the exterior power is simple, a central limit theorem holds for the lift of the (leafwise) hyperbolic Brownian motion to any strongly irreducible, symplectic, -invariant subbundle, that is moreover symplectic-orthogonal to the so-called tautological subbundle. We then show that this implies that a central limit theorem holds for the lift of the Teichmüller geodesic flow to the same bundle. For the random cocycle over the hyperbolic Brownian motion, we prove under the same hypotheses that the variance of the top exponent is strictly positive. For the deterministic cocycle over the Teichmüller geodesic flow we prove that the variance is strictly positive only for the top exponent of the first exterior power (the KZ cocycle itself) under the hypothesis that its Lyapunov spectrum is simple.
Paper Structure (20 sections, 20 theorems, 162 equations)

This paper contains 20 sections, 20 theorems, 162 equations.

Table of Contents

  1. Introduction
  2. Statement of results
  3. Preliminaries
  4. Translation surfaces
  5. Moduli Space
  6. SL(2,R) action
  7. Kontsevich-Zorich cocycle
  8. Hodge inner product and the second fundamental form
  9. Foliated Hyperbolic Laplacian
  10. Harmonic measures
  11. Hyperbolic Brownian Motion
  12. Proofs of Main Theorems
  13. Proof of the Distributional Convergence in Theorem \ref{['rclt']}
  14. Proof of the Distributional Convergence in Theorem \ref{['dclt']}
  15. Positivity of the variance
  16. ...and 5 more sections

Key Result

Theorem 2.1

Let $\text{\bf{H}}$ be a strongly irreducible, symplectic, $\mathrm{SL}(2,\mathbb R)$-invariant subbundle, which is symplectic orthogonal to the tautological subbundle. If $\lambda_k > \lambda_{k+1}$, then there exists a real number $V^{(k)}_{g_\infty} \geq 0$ such that Moreover, if the Lyapunov spectrum is simple, then $V^{(1)}_{g_\infty} > 0$.

Theorems & Definitions (51)

  • Theorem 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Corollary 2.5
  • Theorem 2.6
  • Remark 2.7
  • Remark 3.1
  • Remark 3.2
  • Remark 3.3
  • ...and 41 more