Quid of complex dynamics (in one variable)?
Complex dynamics is the study of the iteration of a holomorphic map from a complex manifold to itself. In this talk, we will focus on the iteration of rational functions from the Riemann sphere to itself.
The objective is to introduce the main objects of study in complex dynamics, particularly the so-called Fatou and Julia sets. After a brief historical introduction, we will define the Fatou and Julia sets of a rational map and present a series of properties they satisfy. We will cover many results, from topological descriptions to equidistribution results.
In the last part of the talk, we will focus on the theory of parabolic implosion. We will use it to prove that the Julia set does not depend continuously on the parameter and to find Julia sets of Hausdorff dimension 2.
