STRUCTURE OF K-INTERVAL EXCHANGE TRANSFORMATIONS: INDUCTION, TRAJECTORIES, AND DISTANCE THEOREMS By S´ EBASTIEN FERENCZI AND LUCA Q. ZAMBONI Abstract. We define a new induction algorithm for k-interval exchange trans- formations associated to the “symmetric" permutation i → k - i +1. Acting as a multi-dimensional continued fraction algorithm, it defines a sequence of gen- eralized partial quotients given by an infinite path in a graph whose vertices, or states, are certain trees we call trees of relations. This induction is self-dual for the duality between the usual Rauzy induction and the da Rocha induction. We use it to describe those words obtained by coding orbits of points under a symmetric interval exchange, in terms of the generalized partial quotients associated with the vector of lengths of the k intervals. As a consequence, we improve a bound of Boshernitzan in a generalization of the three-distances theorem for rotations. However, a variant of our algorithm, applied to a class of interval exchange trans- formations with a different permutation, shows that the former bound is optimal outside the hyperelliptic class of permutations. 1 Preliminaries Interval exchange transformations were originally introduced by Oseledec [27], following an idea of Arnold [2]; see also Katok and Stepin [20]. An exchange of k intervals, denoted throughout this paper by I , is given by a probability vector of k lengths (α 1 ,...,α k ) together with a permutation π on k letters. The unit in- terval is partitioned into k subintervals of lengths α 1 ,...,α k which are rearranged by I according to π. It was Rauzy [29] who first saw interval exchange trans- formations as a possible framework for generalizing the well-known interaction between circle rotations on one hand and Sturmian sequences on the other via the continued fraction algorithm (see, for example, the survey [7]). He defined an algorithm of renormalization for interval exchange transformations, now called Rauzy induction, which generalizes the Euclid algorithm and coincides with it for k =2. JOURNAL D’ANALYSE MATH ´ EMATIQUE, Vol. DOI 1