Orbit structure of interval exchange transformations with flip

A sharp bound on the number of invariant components of an interval exchange transformation (IET) 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) + 2n(min) <= n. Moreover, it is shown that almost all IETs are typical, that is, all have stable periodic components and all the minimal components are 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 IET of n intervals.