A rational family of singular perturbations. The trichotomy theorem.

Abstract

[en] The main focus of this paper will be studying the so called Escape Trichotomy for the singular perturbation family of functions. However, to be able to understand it, there will first be necessary to know some concepts and results about complex dynamical systems and more particularly, the asymptotic behaviour of rational maps on the complex sphere. First, we will introduce the fundamental dynamical systems concepts, such as orbits, fixed and periodic points, attracting and repelling cycles and so on. Then, we will introduce a simpler way of studying a dynamical systems, conjugacies, by establishing if its behaviour under iterations may be similar to another, better known one. We will continue by introducing the concept of critical points and why they are a major point of interest in dynamics.