On The Set Of Periodic Orbits For Interval Exchange Transformations With Flips

This work study the dynamics of interval exchange transformations (IET) with flips. The IET is a class of one-dimensional bijective discontinuous maps that preserve the Lebesgue measure. These maps appear as the Poincare maps for some classes of billiards or flows on surfaces. In the case of an IET which preserve the orientation (an IET without flips), it is known that almost every exchange transformation is ergodic, i.e. it has a unique probability invariant measure. The situation changes when the map reverses orientation of at least one of the intervals, i.e. the map is given by a monotonically decreasing function on this interval. In this case almost every IET with flips has a periodic orbit and nonergodic. This paper study the set of the periodic orbits for an IET with flips. One of the results of this work is the fact that all trajectories are periodic in the case of two and three intervals. Also we prove that an exchange transformation of n intervals with flips has no more than n periodic orbits with the negative multiplier. The main tool to prove the results is the generalization of the Rauzy renormalization which is effectively used for the study of IETs without flips.