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.

Acknowledgements

The careful comments of one referee helped us to improve the presentation and the bibliography, and to correct some imprecisions. This research was supported by EPSRC grant No. GR/S62802/01.

Notes

†If G is a directed graph G with vertex set V and edge set E, the derived graph D(G) has vertex set E, and there is an edge from e∈ E to f∈ E if and only if e and f are consecutive in G, i.e., if the target vertex of e is the source of f in G.

†With apologies to Gertrude Stein, a ρ is a ρ is a ρ.