Orbit structure of interval exchange transformations with flip
Abstract
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 nper and the number of minimal components nmin of an interval exchange transformation of n intervals satisfy nper+2 nmin 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 (nper, nmin) for all typical interval exchange transformation of n intervals.
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