Structure of Three-interval Exchange Transformations Ii: a Combinatorial Description of the Trajectories

We describe an algorithm for generating the symbolic sequences which code the orbits of points under an interval exchange transformation on three intervals. The algorithm has two components. The first is an arithmetic division algorithm applied to the lengths of the intervals. This arithmetic construction was originally introduced by the authors in an earlier paper and may be viewed as a two-dimensional generalization of the regular continued fraction. The second component is a combinatorial algorithm which generates the bispecial factors of the associated symbolic subshift as a function of the arithmetic expansion. As a consequence we obtain a complete characterization of those sequences of block complexity 2n + 1 which are natural codings of orbits of three-interval exchange transformations, thereby answering an old question of Rauzy.