course: Chaoticity of the Teichmuller flow

The first application is the study of the Zorich phenomenon, regarding the asymptotic behavior in homology of trajectories of the vertical flow on a typical translation surface. To first approximation, trajectories grow linearly along the direction of the Schwartzmann asymptotic cycle. Closer inspection (first observed numerically) reveals a hierarchy of polynomial deviations organized in a Lagrangian flag: there is a main deviation concentrated along a 2-plane, followed by a secondary deviation concentrated along a 3-plane, etc. The work of Kontsevich and Zorich explained how this phenomenon is linked to the Lyapunov exponents of a certain linear skew-product over the Teichmuller flow, the Kontsevich-Zorich cocycle. The emergence of the Lagrangian flag was shown to be a consequence of the Zorich-Kontsevich conjecture, that the Lyapunov spectrum of this cocycle is simple. We will discuss an approach to this conjecture based on treating the Kontsevich-Zorich cocycle as a random matrix product. Simplicity of the spectrum is then a consequence of ''combinatorial richness'' of the cocycle, detected by hyperbolichyperbolicboundary behavior.