Limit theorems and the Kontsevich-Zorich Cocycle

Abstract

This thesis concerns the study of Teichmuller dynamics, and in particular the Kontsevich-Zorich cocycle, which is a central object in that theory. In particular, we study and prove limit theorems for this cocycle. In Chapter 2, we present a mechanism for producing oscillations along the lift of the Teichmuller geodesic flow to the (real) Hodge bundle, as the basepoint surface is deformed by a unipotent element of SL(2,R). We apply our methods to all connected components of strata which exhibit a varying phenomenon for the sum of non-negative Lyapunov exponents. In genus 4, by work of Chen-Moller, eight connected connected components of strata exhibit varying Lyapunov exponents, and so we apply our methods to those connected components that are shown to be varying by their work. In Chapter 3, which can be read independently of Chapter 2, we show that a central limit theorem holds for the top exterior power of the Kontsevich-Zorich cocycle. In particular, we show that a central limit theorem holds for the the lift of the (leafwise) hyperbolic Brownian motion to the Hodge bundle, and then show that a (possibly degenerate) central limit theorem holds for the the lift of the Teichmuller geodesic flow to the same bundle. We show that the variance of the limiting distribution for the random cocycle is positive if the second top Lyapunov exponent of the cocycle is positive.