Structure of three-interval exchange transformations II: a combinatorial description of the tranjectories

Ferenczi, Sébastien; Holton, Charles; Zamboni, Luca Q.

doi:10.1007/bf02893083

Cited by 50 publications

(66 citation statements)

References 19 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…In this paper, carrying over the results of an earlier work of the authors together with C. Holton [13], from k = 3 to the much more involved case of 4 intervals STRUCTURE OF K-IETS 3 and more, we accomplish what we wanted for the class of interval exchange transformations where the permutation is the "symmetric" one σ(i) = k + 1 − i for all 1 ≤ i ≤ k, and where the discontinuities β i of I −1 and γ i of I satisfy the inequalities β 1 < γ 1 < β 2 < γ 2 < · · · < β k < γ k (in this case we say I is a symmetric k-interval exchange transformation with alternate discontinuities); this corresponds to a special case of the hyperelliptic stratum. For this class, we define a new induction algorithm, where at each stage we induce I on a disjoint union of k − 1 intervals, each one containing a discontinuity β i and having its extremities on orbits of the discontinuities γ j .…”

Section: Preliminariesmentioning

confidence: 99%

“…This can be done from the present algorithm, by computing return words as in [13] (in all cases for k = 3) or [15] (in examples for k = 4), but, though it is possible in each given example, a general formula seems out of reach for k > 3; from the classical algorithms, no general formula is available either.…”

Section: Substitutions Definition 34mentioning

confidence: 99%

“…A difficult open question is to prove that a k-interval exchange transformation corresponds to a substitutive system if and only if the lengths of the intervals are algebraic of degree at most k. The "if" part is known, and could be deduced from the above result by using substitutions on k letters, but the "only if" part has been proved only for k = 3 [13], and in that case the degree always falls to k − 1 = 2.…”

Section: Substitutions Definition 34mentioning

confidence: 99%

See 2 more Smart Citations

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

Ferenczi

Zamboni

2010

JAMA

Self Cite

View full text Add to dashboard Cite

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...

show abstract

Section: Preliminariesmentioning

confidence: 99%

Section: Substitutions Definition 34mentioning

confidence: 99%

Section: Substitutions Definition 34mentioning

confidence: 99%

See 1 more Smart Citation

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

Ferenczi

Zamboni

2010

JAMA

Self Cite

View full text Add to dashboard Cite

show abstract

“…Then the word u ρ is aperiodic, uniformly recurrent, and the language of u ρ does not depend on the intercept ρ. The complexity of the infinite word u ρ is known to satisfy C uρ (n) ≤ 2n + 1 (see [6]). If for every n ∈ N equality is achieved, then the transformation T and the word u ρ are said to be non-degenerate.…”

Section: Exchange Of Three Intervalsmentioning

confidence: 99%

“…These specific cases are studied in Section 5. This allows us to describe the return words to palindromic bispecial factors, which can be seen as a complement to some of the results in [6]. We also focus on substitutions fixing words coding interval exchange transformations.…”

Section: Introductionmentioning

confidence: 99%