Dinámica de la familia exponencial compleja

Abstract

[en] The aim of this project is to study the dynamics of the complex exponential family $E_{\lambda}(z), \lambda \in \mathbb{C}$. In the first instance, we are going to explain the background of complex anaylisis and holomorphic dynamics, specially in trascendental entire functions. Then, we will expose the behaviour of the function $e^{z}$ and we are going to study the fixed points of $E_{\lambda}$ for $\lambda \in \mathbb{R}$. Finally, we are going to define, in general terms, the Julia and Fatou sets respectively and prove two results of M. Misiurewicz (1980) and R. Devaney (1994) respectively,. The first result shows that $\mathcal{J}\left(E_{1}\right)=\mathbb{C}$ and, for this reason, $\mathcal{F}\left(E_{1}\right)=\emptyset$. The second result shows that $\exists\left\{\lambda_{n}\right\}_{n \in \mathbb{N}} \underset{n \rightarrow \infty}{\longrightarrow} 1$ such that $\mathcal{F}\left(E_{\lambda_{n}}\right) \neq \emptyset$ for all $n \in \mathbb{N}$.