A left continuous, piecewise translation of the interval [0,1) is called an interval exchange transformation. We would prove that the number of discontinuity points of such a map grows either linearly or stays bounded. The geometric prove of that fact will also show the if that number stays bounded then there is an obvious reason for that. As a corollary we would discuss various properties of the group of interval exchange transformations.

The talk will be based on two papers: