It\'eration d'applications rationnelles dans les espaces de matrices

TL;DR

This paper investigates the dynamics of rational maps on matrix spaces, focusing on maps like squaring matrices and their perturbations, extending understanding from one-dimensional cases to higher dimensions.

Contribution

It introduces new properties of rational maps on matrix spaces and analyzes their dynamical behavior, including perturbations, in higher-dimensional settings.

Findings

Analysis of the map M -> M^2 and its dynamical properties

Characterization of weak complexity in matrix rational maps

Behavior of small perturbations of the squaring map

Abstract

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.