Unique Ergodicity of Circle and Interval Exchange Transformations with Flips

We study circle exchange transformations on three subintervals, that reverse orientation on at least one of the subintervals (a flip). We prove that if such a transformation has exactly one flip, then it has a periodic orbit and thus no dense orbit. For the cases of two and three flips, we construct uniquely ergodic minimal examples by finding a periodic point of an adapted Rauzy–Veech renormalisation operator. Finally, by means of an inductive reasoning, we show that for any n ≥ 4 and 1 ≤ f ≤ n, there exist uniquely ergodic transitive circle exchange transformations on n subintervals with f flips.