On the Brin Prize work of Artur Avila in Teichmüller dynamics and interval-exchange transformations (Q456679)

This paper reviews the work of Artur Avila and his collaborators (notably G. Forni, S. Gouezel, M. Viana, and J.-C. Yoccoz) on the ergodic properties of interval exchange transformations (IETs) and related problems on the dynamics of the Teichmüller (geodesic) flow. It surveys the connection between IETs and the Teichmüller flow, especially the link between exponents of deviation of ergodic averages for IETS and Lyapunov exponents for the Kontsevich-Zorich cocycle over the Teichmüller flow, and explains (with outlines of proofs) the results of Avila-Viana on simplicity of the Lyapunov spectrum, and Avila-Gouezel-Yoccoz on exponential rates of mixing for the Teichmüller flow. It also discusses his proof (joint with G. Forni) of weak-mixing of almost every interval exchange transformation. It is an eminently readable survey of an impressive sample of Avila's work.