New points of view in rational dynamics in several variables – Fatou

FATOU is a project in pure mathematics, in the field of dynamical systems. Its purpose is the study of phase space and parameter spaces of holomorphic dynamical systems in several complex variables. Recent developpements in the field suggest that those dynamical systems can have fundamentally different properties than holomorphic dynamical systems in one complex variable, such as wandering Fatou components or open sets of bifurcations in parameter spaces. The ambition of this project is to consider these problems under a new scope.


Classically, both the phase and parameter spaces can be described as the union of a stable part and a chaotic part. It has been a major problem in modern dynamics to study to precisely how we can dynamicaly describe this dichotomy.
In the context of holomorphic dynamical systems, the description of both parameter and dynamical spaces is somehow more tricky. As examples, one may cite for example the filtration of the chaotic part of the dynamics of an endomorphism f of P^k given by its Green currents and, similarly, the filtration of the set of rational maps of the Riemann sphere which are not structurally-stable on their Julia set induced by the bifurcation currents.

The main idea of this project is to gather young researchers from dynamics in several complex variables, and from parameter spaces in one complex variable dynamics to study both the topological dynamics of holomorphic and birational maps of P^k and parameter space phenomena for holomorphic families of rational maps of P^k. More precisely, we wish to :
- study the structure of the omega-limit set of wandering Fatou components of endomorphisms of P^k and identify sufficient arithmetic or metric conditions ensuring the absence of wandering domains,
- enlight unexpectedly sophisticated dynamical behavior for birational maps,
- understand new phenomena responsible for open sets bifurcations and those responsible for stronger bifurcation phenomena in families of endomorphisms of P^k.

The parent of the project is Amiens and the team consists in 7 young researchers having permanent positions and who are active in the field.

Another objective of the project is to undertake actions promoting this field of research among a general audience (in particular university students) and to promote a fast homogenization of mathematical knowledge inside the group. For this, we will organize two internal meetings every year, which also will help us stay informed about the state of the art, and report about work in progress by the members. An international conference will be organized at the end of the project to review the most recent advances in the topic.

The acronym FATOU was chosen as a reference to Pierre Fatou, one of the historical important researchers in the field in the early twentieth century.