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We consider the restriction of interval exchange transformations to algebraicnumber fields, which leads to maps on lattices. We characterizerenormalizability arithmetically, and study its relationships with ageometrical quantity that we call the drift vector. We exhibit some examples ofrenormalizable interval exchange maps with zero and non-zero drift vector, andcarry out some investigations of their properties. In particular, we look forevidence of the finite decomposition property: each lattice is the union offinitely many orbits.

Geometric representation of interval exchange maps over algebraic number fields