Local connectivity of Julia sets

Abstract

[en] Complex dynamics is a part of Mathematics that did not start to shine until the arrival of prominent figures like Koenigs, Fatou and Julia. In particular, one of the most innovative ideas was the Julia set of a given function f . The particular shape and characteristics of these sets do not leave any mathematician indifferent, and a useful way to try to understand them is to study their topology. We aim to determine which Julia sets are locally connected, considering the relations that this topological property has with major questions of complex dynamics, as for example the MLC conjecture. In this thesis we will focus on hyperbolic rational maps, the maps that have the simplest dynamics, and which are conjectured to be dense among all rational maps (HD conjecture). The goal is to prove the following theorem: hyperbolic rational maps of degree larger than 1 with a connected Julia set have a locally connected Julia set. To do so, we first present the preliminaries on different aspects of Mathematics, such as hyperbolic geometry, Montel’s theory and, of course, complex dynamics. It is followed by the proof of Carathéodory’s theorem, which gives a crucial criterion about which sets are locally connected. Finally, the last chapter is dedicated to the proof of the main theorem.