On Birkhoff Sums and Roth Type Conditions for Interval Exchange Transformations

Understanding growth and behaviour of Birkhoff sums is one of the central themes in the study of interval exchange transformations (IET), starting from the work of Zorich and Forni on deviations of ergodic averages, up to recent results on limit theorems by Bufetov and others. J. C. Yoccoz, in joint work with Marmi and Moussa, introduced an object called "limit shape" which can be used as a tool to study Birkhoff sums which display polynomial deviations. In joint work with Yoccoz and Marmi, we describe a limit object for Birkhoff sumsof functions which correspond to relative homology classes and, as in the case of the circle, display slower growth. We will also discuss two related variations of the Roth type Diophantine condition introduced by Marmi, Moussa and Yoccoz to solve the cohomological equation for IETs. This talk is based on joint work with Jean-Christophe Yoccoz and Stefano Marmi.