  A sharp bound on the number of invariant components of an interval exchange
transformation is provided. More precisely, it is proved that the number of
periodic components n_per and the number of minimal components n_min of an
interval exchange transformation of n intervals satisfy n_per+2 n_min\le n.
Besides, it is shown that almost all interval exchange transformations are
typical, that is, have all the periodic components stable and all the minimal
components robust (i.e. persistent under almost all small perturbations).
Finally, we find all the possible values for the integer vector (n_per, n_min)
for all typical interval exchange transformation of n intervals.

Dynamical Systems