A SELF-DUAL INDUCTION FOR THREE-INTERVAL EXCHANGE TRANSFORMATIONS

Abstract. We describe completely a new induction process on three-interval exchange transformations, under its additive and multiplicative forms; we describe the natural ex-tension of the induction map, show that it is self-dual for the notion of duality which links the Rauzy/Zorich and da Rocha inductions, and compute a finite ergodic invariant measure (for the multiplicative form). Then we explain the connection with the published negative slope algorithm of Ferenczi, Holton and Zamboni, and use our measure-theoretic study to get new results on three-interval exchange transformations. The classical Euclid algorithm of continued fraction approximation for an irrational α can be seen as a renormalization process on the dynamics of the rotation of angle α; this has several forms: the additive form is usually given by the induction map on two coordinates, (x, y) → (x − y, y) if x> y, (x, y) → (x, y − x) if x < y; the multiplicative form, which is an acceleration of the additive one, is given by the Gauss map x → { 1 x}. A useful generalization of this process to simultaneous approximation of several numbers is obtained by replacing rotations by interval exchange transformations, see for example the survey [2]; the additiv