Iteration of transcendental functions

Abstract

[en] In this project we analyze the behavior of transcendental functions under iteration i.e., those with an essential singularity at $\infty$. We emphasize the general case of meromorphic transcendental functions with the aim of understanding the dynamical consequences of the presence of poles.

Finally, we apply these results and techniques to study, on the one hand, the dynamics of the exponential family $E_{\lambda}(z)=\lambda e^{z}$, and on the other hand, the family of meromorphic maps $$ f_{\lambda}(z)=\lambda\left(\frac{e^{z}}{z+1}-1\right). $$ In this last part, which is original work, we prove that under certain conditions, the basin of attraction of $z=0$ is infinitely connected.