El conjunt de Mandelbrot: hiperbolicitat i connectivitat local

Abstract

[en] In this project, we study the behaviour of holomorphic functions of one complex variable under iteration, both locally and globally. We do so by reviewing the principal results that shape the so-called holomorphic dynamics, with emphasis on polynomial maps. The aim is to establish the basis to study the quadratic family $$ \mathcal{Q}:=\left\{P_c(z)=z^2+c \mid c \in \mathbb{C}\right\} $$ We characterize the parameter's c-plane and define the Mandelbrot set: A compact, connected and simply connected set which hides striking properties profoundly related with many other branches of Mathematics. In the last section we comment the principal conjectures which remain unanswered for several decades: the "Mandelbrot's Local Connectivity Conjecture" and the "Density of Hiperbolicity Conjecture".