  The iteration of rational maps is well-understood in dimension 1 but less so
in higher dimensions. We study some maps on spaces of matrices which present a
weak complexity with respect to the ring structure. First we give some
properties of certain rational maps; the simplest example is the rational map
which sends the matrix $\mathrm{M}$ onto $\mathrm{M}^2$ for which we exhibit
some dynamical properties. Finally we deal with some small perturbations of
this map.
