Piecewise isometries have zero topological entropy

We show that piecewise isometries, i.e. non-necessarily invertible maps def ined on a finite union of polytopes and coinciding with an isometry on the interior of each polytope, have zero topological entropy in any dimension. This had been conjectured by a number of authors. The proof is by an induct ion on the dimension and uses a device (the differential of a piecewise lin ear map) introduced by M. Tsujii.