A self-dual induction for three-interval exchange transformations

摘要

We describe completely a new induction process on three-interval exchange transformations, under its additive and multiplicative forms; we describe the natural extension 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 [S. Ferenczi, C. Holton, and L. Zamboni, The structure of three-interval exchange transformations I: An arithmetic study, Ann. Inst. Fourier (Grenoble), 51 (2001), pp. 861-901], and use our measure-theoretic study to get new results on three-interval exchange transformations.