Limit Theorems for the Kontsevich-Zorich Cocycle on Moduli Spaces of Abelian Differentials
The Kontsevich-Zorich cocycle is a fundamental object in the study of the chaotic dynamics of the Teichmueller geodesic flow on moduli spaces of hyperbolic surfaces. A dynamical form of the Law of Large
Numbers shows that the exponential growth rate of the norm of the cocycle along typical orbits converges to its space average. In this talk, we show that these growth rates also obey a Central Limit Theorem. This provides one of the first instances in which such a result holds for deterministic cocycles over flows and generalizes a long history of results in the study of random walks on groups. The main ingredient is a spectral gap result which is established via the theory of anisotropic Banach spaces. No background in Teichmueller theory or spectral theory will be assumed.