Journal d'Analyse Mathématique

, Volume 112, Issue 1, pp 289–328 | Cite as

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

Article

Abstract

We define a new induction algorithm for k-interval exchange transformations associated to the “symmetric” permutation iki + 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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Copyright information

© Hebrew University Magnes Press 2010

Authors and Affiliations

  1. 1.Institut de Matháematiques de LuminyCNRS - UMR 6206 CASE 907Marseille Cedex 9France
  2. 2.Fédération de Recherche des Unités de Mathématiques de MarseilleCNRS - FRParisFrance
  3. 3.Institut Camille JordanUniversité Claude Bernard Lyon 1Villeurbanne CedexFrance
  4. 4.School of Computer ScienceReykjavik University103 ReykjavikIceland

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