Languages of <i>k</i>
-interval exchange transformations

Ferenczi, Sébastien; Zamboni, Luca Q.

doi:10.1112/blms/bdn051

Cited by 47 publications

(78 citation statements)

References 15 publications

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“…The proof of (ii) is similar to the proof of Theorem 2.9, with Proposition 7 of [16] replaced by the more general Theorem 2 of the same paper.…”

Section: Proof (I) Is the Construction Abovementioning

confidence: 69%

“…Given an infinite path, it follows from the above proof and [16] that a vector of length for an associated I is given by (µ[1], . .…”

Section: Proofmentioning

confidence: 99%

“…Now, Proposition 7 of [16] states that a language L is the language of a symmetric k-interval exchange transformation with alternate discontinuities, satisfying the i.d.o.c. condition, if (and only if) it is minimal, the set of words of length 2 is {1k, 1(k − 1), 2(k − 1), 2(k − 2), .…”

Section: Proofmentioning

confidence: 99%

Section: Preliminariesmentioning

confidence: 99%

“…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 intersects this initial sgement, and then considers shorter and shorter initial segments.…”

mentioning

confidence: 99%

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

Ferenczi

Zamboni

2010

JAMA

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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...

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“…The proof of (ii) is similar to the proof of Theorem 2.9, with Proposition 7 of [16] replaced by the more general Theorem 2 of the same paper.…”

Section: Proof (I) Is the Construction Abovementioning

confidence: 69%

“…Given an infinite path, it follows from the above proof and [16] that a vector of length for an associated I is given by (µ[1], . .…”

Section: Proofmentioning

confidence: 99%

Section: Proofmentioning

confidence: 99%

Section: Preliminariesmentioning

confidence: 99%

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