Structure of K-interval exchange transformations: Induction, trajectories, and distance theorems

Abstract

Abstract. We define a new induction algorithm for k-interval exchange transformations associated to the "symmetric" permutation i → k − i + 1. Acting as a multi-dimensional continued fraction algorithm, it defines a sequence of generalized 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 transformations with a different permutation, shows that the former bound is optimal outside the hyperelliptic class of permutations. PreliminariesInterval 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 interval 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 transformations 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] [4]. But, in contrast with the Sturmian case, though this induction gives access to the symbolic dynamics of the trajectories, it is difficult to describe them explicitly and indeed the Rauzy induction was never actually used for that purpose. A partial description of the trajectories was given in [24], and it aroused interest by defining a different kind of induction; this da Rocha induction was then considered to be dual to the Rauzy induction: the Rauzy induction builds the points where the orbits of the discontinuities of I approximate 0, while the da Rocha induction builds the points where the orbit of 0 approximates the discontinuities of I −1 . Note that (non-constructive) combinatorial characterizations of the language of trajectories were given recently in [16] and [5].These inductions are also a fundamental tool in the study of the space of moduli of Riemann surfaces, and the various strata of its unit tangent bundle, through what has been a basic object of interest for the last 25 years, the Teichmüller flow on a stratum; in the case of interval exchanges, we consider the Teichmüller flow for the Riemann surface obtained by glueing parallel opposite sides of a polygon. The Rauzy induction chooses an initial segment of a horizontal separatrix and follows its vertical separatrix till it i...

A self-dual induction for three-interval exchange transformations

“…The Rokhlin towers constitute a discrete form of the rectangles defining Σ; in the symbolic vocabulary of [8], the heights of the Rokhlin towers correspond to the lengths of the prefixes.…”

“…A new induction algorithm is defined for all k by Ferenczi and Zamboni [8] (see also [9] for k = 4); this exists as yet only under an additive form, and its connection with the negative slope algorithm is not explicit. The present paper aims to fill this gap and to study completely the new algorithm for k = 3: we first discuss its definition, then give a self-contained description of the induction process under its additive form, and deduce its multiplicative form; we identify the natural extension of this induction process, and show that it is self-dual for the Rauzy/da Rocha duality mentioned above.…”

“…The motivation for this is to get more information on the dynamics and combinatorics of interval-exchange transformations, and that has indeed been done in [5][6] [7] for 3 intervals, using a variant of this algorithm, see section 4 below, and in [8][9] for k intervals. The reasons why this new algorithm should give informations which the existing induction algorithms could not give are twofold:…”

“…• by approximating the discontinuities of T −1 from the right and the left, we build small intervals around them, and these correspond to the successive bispecial factors of the language of T , see [8] for the word-combinatorial aspects of this definition.…”

“…Our algorithm is a particular (and more explicit) case of the one defined in [8], for k intervals. Note that exchanges of two intervals are just rotations of angle α; our algorithm for two intervals would find the iterates of the point 1 − α which approximate the point α, from the right or from the left; as there is only one interval to cut at each stage, there is no choice involved, and what we retrieve is just the additive Euclid algortihm.…”