Discretizations of isometries

This paper deals with the dynamics of discretizations of isometries of ${\mathbf{R}}^n$, and more precisely the density of the successive images of ${\mathbf{Z}}^n$ by the discretizations of a generic sequence of isometries. We show that this density tends to 0 as the time goes to infinity. Thus, in general, all the information of a numerical image will be lost by applying many times a naive algorithm of rotation.