abstract: This course is a follow-up to Yoccoz's course. It will center around applications of the "chaotic" aspect of the dynamics of the Teichmuller flowRenormalization algorithm for interval exchange transformations.
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.
The second problem is the weak mixing property for typical interval exchange transformations which are not rotations. Here what one has to show is absence of measurable eigenfunctions. The renormalization algorithm ''smooths'' eigenfunctions, making them approximately constant (this is a common theme of renormalization theory). This leads to the Veech criterium for ruling out eigenvalues, in terms of the dynamics of a linear skew-product (the Kontsevich-Zorich cocycle), acting as automorphisms of tori. Chaoticity allows us to proceed by stochastic modelling. The analysis involves some information on the Lyapunov exponents and combinatorial richness (strictly contained in the discussion of the previous problem).