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

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Astract

We describe an algorithm for generating the symbolic sequences which code the orbits of points under an interval exchange transformation on three intervals. The algorithm has two components. The first is an arithmetic division algorithm applied to the lengths of the intervals. This arithmetic construction was originally introduced by the authors in an earlier paper and may be viewed as a two-dimensional generalization of the regular continued fraction. The second component is a combinatorial algorithm which generates the bispecial factors of the associated symbolic subshift as a function of the arithmetic expansion. As a consequence, we obtain a complete characterization of those sequences of block complexity 2n+1 which are natural codings of orbits of three-interval exchange transformations, thereby answering an old question of Rauzy.

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Authors and Affiliations

  1. Institut de Mathématiques de Luminy, CNRS-UPR 9016, Case 907, 163 av. de Luminy, F13288, Marseille Cedex 9, France

    Sébastien Ferenczi

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

    Sébastien Ferenczi

  3. Department of Mathematics, University of California, 94720-3840, Berkeley, CA, USA

    Charles Holton

  4. Department of Mathematics, University of North Texas, 76203-1430, Denton, TX, USA

    Luca Q. Zamboni

Authors
  1. Sébastien Ferenczi
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  2. Charles Holton
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  3. Luca Q. Zamboni
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Correspondence to Sébastien Ferenczi.

Additional information

Partially supported by NSF grant INT-9726708.

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Ferenczi, S., Holton, C. & Zamboni, L.Q. Structure of three-interval exchange transformations II: a combinatorial description of the tranjectories. J. Anal. Math. 89, 239–276 (2003). https://doi.org/10.1007/BF02893083

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  • DOI: https://doi.org/10.1007/BF02893083

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