A PhD course
Period: 14-23 January 2019
Dipartimento di Matematica

1. Crash course in one-dimensional complex dynamics.
Classification of invariant Fatou components and non-existence of wandering domains. Discussion of fixed points.

2. Parabolic bifurcations.
Mane-Sad-Sullivan result. Lavaurs Theorem.

3. Wandering domains in higher dimensions: skew products.
Detailed proof of the construction of wandering domains. Discussion of very recent results.

4. Fatou components of polynomial automorphisms, and open questions.

Abstract

The iteration of rational functions, acting on the Riemann sphere, is a classical subject that has been studied for well over a century. It is remarkable that even the iteration of quadratic polynomials, a seemingly simple family of maps, has led to a surprisingly deep theory, with several important open questions still unsolved. Naturally the field of holomorphic dynamical systems has expanded into many directions. Since the late 1980's there has been considerable interest in the iteration of rational functions in higher dimensions. In recent years it has become clear that already in two complex variables phenomena arise that are unlike anything that can occur in one variable. In this course we will consider some of the most striking results in this direction of research. In particular we consider the construction of wandering domains, and we finish by discussing the most pressing open questions regarding the classification of invariant Fatou components.