Non-uniquely ergodic interval exchange transformations

Interval exchange transformations (IETs) appeared in 60s as the generalisation of the circle rotations as well as first return maps to the transversal for the billiard flow in a rational polygon. IET was introduced by M. Keane. An IET is called minimal, if all infinite orbits are dense. M. Keane formulated condition for IET minimality and showed that almost all IET are minimal. Then H. Masur and W. Veech respectively proved that almost every IET is uniquely ergodic with respect to Lebesgue measure. Subsequently, J. Chaika obtained estimates of the Hausdorff dimension of invariant measures from the example of M. Keane. In my report, I will talk about non-uniquely ergodic examples of IETs and methods for estimating the Hausdorff dimension of each invariant measures