Tag: Lucas Kaufmann

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.

The field of complex dynamics deals with the study of the iteration of a map from a complex manifold to itself. The one dimensional theory is more than one-hundred years old and is now very well developed.

Due to the fundamental differences between complex analysis in one and higher dimensions, the study of higher dimensional dynamical systems is much more recent and is still an active research domain, with interesting connections to complex geometry, algebraic geometry, number theory, mathematical physics etc.

The aim of this talk is to survey standard results on the dynamics of self-maps in complex manifolds. I will focus on the case of endomorphisms of the complex projective space and use it to illustrate how the tools of pluripotential theory and the theory of currents are crucial in their study.

In particular, I aim to talk about Green currents and measures, equidistribution theorems and, if time permits, relations with the intersection theory of currents.