An interval exchange transformation (IET) is a map from the unit interval to itself for which the interval admits a finite partition by subintervals over each of which, the map acts as translation in such a way that the image of the interior of the intervals is disjoint from one another. This family of maps arise naturally as the return maps of directional flows on translation surfaces. The ergodic theory of IETs has been studied extensively by many authors. In this talk, we will review some ergodic properties of this family. Then, we introduce linear cocycles over the space of IETs and study their spectral properties. A well-known cocycle is the Kontsevich-Zorich cocycle. The study of the Lyapunov spectra of this cocycle has implications for asymptotic growth of Birkhoff sums and understanding of the dynamics of typical IETs. One can show that this cocycle preserves a symplectic structure, thereby having symmetric spectra with respect to zero. The Kontsevich-Zorich conjecture proved by Forni, Avila-Viana asserts that the Lyapunov spectra of the restricted Kontsevich-Zorich cocycle, and consequently that of the restricted Zorich cocycle, are simple and nonzero. Our main focus in this talk, will be on the study of the spectrum of another linear cocycle introduced by Bufetov and Solomyak, known as the twisted (or spectral) cocycle. This cocycle is designed so that its iterations control the exponential Birkhoff sums of locally constant functions, analogous to the fact that the iterations of the Zorich cocycle govern the behavior of ordinary Birkhoff sums of such functions. This cocycle and its cousins have appeared in several papers studying rates of weak mixing and the spectral measures associated with IETs and substitution dynamical systems. We will show that the spectrum of this cocycle is non-degenerate if the genus of the associated translation surface is bigger than one.
This talk is based on a recent joint work with Pedram Safaee (UZH).