On Periods of Interval Exchange Transformations

Abstract

In this paper, we study the periods of interval exchange transformations. First, we characterize the periods of interval exchange transformations with one discontinuity. In particular, we prove that there is no forcing between periods of maps with two branches of continuity. This characterization is a partial solution to a problem by Misiurewicz. In particular, a periodic structure is not possible for a family of the maps with one point of discontinuity in which the monotonicity changes. Second, we study a similar problem for interval exchange transformations with two discontinuities. Here we classify these maps in several classes such that two maps in the same class have the same periods. Finally, we study the set of periods for two categories, obtaining partial results that prove that the characterization of the periods in each class is not an easy problem.