A generalization of the self-dual induction to every interval exchange transformation

Ferenczi, Sébastien

Annales de l'Institut Fourier, Tome 64 (2014) no. 5, pp. 1947-2002.

Nous généralisons à tous les échanges d’intervalles l’algorithme d’induction défini par Ferenczi et Zamboni pour une classe particulière. Chaque échange d’intervalles correspond à un chemin infini dans un graphe dont les sommets sont certaines unions d’arbres que nous appelons des forêts de châteaux. Nous l’utilisons pour décrire les mots obtenus en codant les trajectoires, et donner une représentation explicite du système par des tours de Rokhlin. Comme application, nous construisons le premier exemple connu d’échange d’intervalles faiblement mélangeant en-dehors de la classe de Rauzy hyper-elliptique et de celle des rotations.

We generalize to all interval exchanges the induction algorithm defined by Ferenczi and Zamboni for a particular class. Each interval exchange corresponds to an infinite path in a graph whose vertices are certain unions of trees we call castle forests. We use it to describe those words obtained by coding trajectories and give an explicit representation of the system by Rokhlin towers. As an application, we build the first known example of a weakly mixing interval exchange outside the hyperelliptic and rotations Rauzy classes.

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