EVENT DETAILS AND ABSTRACT

Title: Ergodic Theory of Interval Exchange Transformations
Speaker: Howard Masur
Speaker Info: University of Chicago
Brief Description:
Special Note:
Abstract:

An interval exchange transformation (IET) is a map of an interval to itself defined by cutting the interval into d pieces and rearranging them by translations. IET arise in the study of rational billiards and translation surfaces. An IET with 2 intervals is equivalent to a rotation of a circle. In that case it is classical that there is a dichotomy: if the rotation angle is rational then every orbit is periodic or it is irrational and every orbit is dense and what is stronger; every orbit is equidistributed on the circle; a property called unique ergodicity. Once d is at least 4 there exist examples of IET with dense orbits which are not uniquely ergodic. I will discuss the issue of whether a non ergodic IET may still possess some orbit that is equidistributed. Along the way I will introduce the process of Rauzy induction, a generalization of the continued fraction algorithm, which is one of the main tools in studying IET. This is joint work with Jon Chaika.