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.
Categories: math.DS math.CV