Interval exchange transformations over algebraic number fields: the cubic Arnoux–Yoccoz model

Abstract

We apply methods developed for two-dimensional piecewise isometries to the study of renormalizable interval exchange transformations over an algebraic number field , which lead to dynamics on lattices. We consider the -module generated by the translations of the map. On it, we define an infinite family of discrete vector fields, representing the action of the map over the cosets , which together form an invariant partition of the field . We define a recursive symbolic dynamics, with the property that the eventually periodic sequences coincide with the field elements. We apply this approach to the study of a model introduced by Arnoux and Yoccoz, for which λ−1 is a cubic Pisot number. We show that all cosets of decompose in a highly non-trivial manner into the union of finitely many orbits.