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

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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.

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Authors and Affiliations

  1. Institut de Matháematiques de Luminy, CNRS - UMR 6206 CASE 907, 163 av. de Luminy, F13288, Marseille Cedex 9, France

    Sébastien Ferenczi

  2. Fédération de Recherche des Unités de Mathématiques de Marseille, CNRS - FR, 2291, Paris, France

    Sébastien Ferenczi

  3. Institut Camille Jordan, Université Claude Bernard Lyon 1, 43 Boulevard du 11 novembre 1918, F69622, Villeurbanne Cedex, France

    Luca Q. Zamboni

  4. School of Computer Science, Reykjavik University, Kringlan 1, 103 Reykjavik, Iceland

    Luca Q. Zamboni

Authors
  1. Sébastien Ferenczi
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  2. Luca Q. Zamboni
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Correspondence to Sébastien Ferenczi.

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Ferenczi, S., Zamboni, L.Q. Structure of K-interval exchange transformations: Induction, trajectories, and distance theorems. JAMA 112, 289–328 (2010). https://doi.org/10.1007/s11854-010-0031-2

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  • DOI: https://doi.org/10.1007/s11854-010-0031-2

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