Recurrence properties of the interval exchange map

The Interval exchange map generalizes the irrational rotation of the circle. We discuss Diophantine type of the irrational rotation to the interval exchange map and introduce a Diophantine type for the interval exchange maps using the asymptotic rate of the return time. It is a generalization of the Roth's type condition for the irrational rotation. Also the asymptotic rate of hitting time is considered.