The escaping set

Abstract

The aim of this project is to understand the behaviour of holomorphic functions of one complex variable under iteration, focusing on polynomials and transcendental entire functions. Our study centres on the points whose orbits tend to infinity, which form the escaping set, a fundamental object in complex dynamics. The escaping set provides insight into the global behaviour of iterates and their relationship with the Julia and Fatou sets, which are also important sets in complex dynamics. To achieve this, we begin by establishing a foundational background in dynamical systems. We then proceed with a separate study to characterize the escaping set for both polynomials and transcendental entire functions, using the previous dynamical results as tools to analyse their structure and properties. In both cases, the most remarkable result is that the escaping set is nonempty, proving the existence of points whose orbits eventually escape to infinity under iteration.