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- COMBINATORIAL METHODS FOR INTERVAL EXCHANGE TRANSFORMATIONS hal link

Auteur(s): Ferenczi Sébastien

(Article) Publié: Southeast Asian Bulletin Of Mathematics, vol. 37 p.47-66 (2013)


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Résumé:

This is a survey on the big questions about interval exchanges (minimality, unique ergodicity, weak mixing, simplicity) with emphasis on how they can be tackled by mainly combina-torial methods. Interval exchange transformations, defined in Definition 1 below, constitute a famous class of dynamical systems; they were introduced by V. Oseledec [25], and have been extensively studied by many famous authors; up to now, the main results in this swifly-evolving field can be found in the two excellent courses [33] and [34]. To study interval exchanges, three kind of methods can be used: by definition, these systems are one-dimensional, and the first results on them naturally used one-dimensional techniques; then the strongest results on interval exchanges have been obtained by lifting the transformation to higher dimensions and using deep geometric methods. However, many of these results have been reproved by using zero-dimensional methods; these use the codings of orbits to replace the original dynamical system by a symbolic dynamical system, as in Definition 4 below. Now, most of the existing texts, including the two courses mentioned above, focus on the geometric methods; the present survey wants to emphasize what can be achieved by the two other kinds of methods, which have both a strong flavour of combinatorics. The one-dimensional methods yield the basic results, some of which the reader will find in Section 2 below, but also the famous Keane counterexamples described in Section 4, and a very nice new result of M. Bosher-nitzan which is the object of our Section 6; Sections 3 and 5 are devoted to the zero-dimensional methods; the necessary definitions of word combinatorics, symbolic and measurable dynamics are given in Section 1. All those sections are also retracing the colourful history of the theory of interval exchanges, made with big conjectures brilliantly solved after long waits; thus we finish the paper by explaining in Section 7 the last big open question in the domain. This paper stems from a course given during the summer school Dynamique en Cornouaille, which took place in Fouesnant in june 2011; the author is very grateful to the organizer, R. Lep-laideur, for having commandeered it.