Fagella Rabionet, NúriaPujol Vidal, Àlex2023-03-222023-03-222022-06-13https://hdl.handle.net/2445/195766Treballs Finals de Grau de Matemàtiques, Facultat de Matemàtiques, Universitat de Barcelona, Any: 2022, Director: Núria Fagella Rabionet[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".68 p.application/pdfcatcc-by-nc-nd (c) Àlex Pujol Vidal, 2022http://creativecommons.org/licenses/by-nc-nd/3.0/es/Sistemes dinàmics hiperbòlicsTreballs de fi de grauFuncions de variables complexesFuncions meromorfesHyperbolic dynamical systemsBachelor's thesesFunctions of complex variablesMeromorphic functionsinfo:eu-repo/semantics/bachelorThesisinfo:eu-repo/semantics/openAccess