Approximation of Baker domains and convergence of Julia sets.

Abstract

The goal of this thesis is to prove the Hausdorff convergence of Julia sets as we approximate a family of transcendental entire functions featuring a unique Baker domain. At first, we give a dynamical description of the approximating transcendental functions and show the existence of invariant structures in the Fatou set under iterates. In particular, the approximating functions have a basin of attraction converging to the Baker domain as kernels in the sense of Carathéodory. Finally, we prove Hausdorff convergence in two different ways. On the one hand, given certain conditions on the Fatou set of the limit functions we obtain Hausdorff convergence of the Julia sets. On the other hand, using different conditions on the Fatou set we obtain convergence of the filled Julia sets, which are defined with respect to the Baker domains or the approximating basin of attraction.