The study of functional equations is at the origin of the early developments of the iteration theory of polynomials and rational functions, carried out by Fatou, Julia, Ritt and others. Among these equations, the commutation relation f g = g f is particularly interesting. In this talk I will discuss the following problem: can we classify commuting pairs of holomorphic endomorphisms of the complex projective space?
We will see that the relation f g = g f gives rise to special symmetries of several dynamical objects attached to these maps, such as their invariant currents and measures, their Julia sets and so on. This rigidity allows us to understand the structure of these maps and even give a full description in low dimensions.