Geometric representation of interval exchange maps over algebraic number fields

This paper is concerned with the restriction of interval exchange transformations (IETs) to algebraic number fields, which leads to maps on lattices. We characterize renormalizability arithmetically and study its relationships with a geometrical quantity that we call the drift vector. We exhibit some examples of renormalizable IETs with zero and non-zero drift vectors and carry out some investigations of their properties. In particular we look for evidence of the finite decomposition property on a family of IETs extending the example studied in Lowenstein et al (2007 Dyn. Syst. 22 73–106).