We consider the family of transcendental entire functions given by {f c : C → C : z − c + e z , c ∈ C}. If Re c > 0, then f c features a Baker domain as the only component of the Fatou set, while the functions f c show a different dynamical behaviour if c ∈ iR. We describe the dynamical planes of these functions and show that the Julia sets converge in the limit process f c 1 +ic 2 → f ic 2 with respect to the Hausdorff metric, where c 1 ∈ R + and c 2 ∈ R. We use this to show that Baker domains of any type (concerning a classification of König) are not necessarily stable under perturbation.

for many very fruitful discussions. I was staying at universities in Barcelona, Paris, Mexico and Rio de Janeiro, thanks to everyone who made this possible. Thanks to everyone I had the pleasure to work with, and especially to Jordi and Toni for valuable help on the programming issues. Finally, thanks to Mum and Dad. This reseach was funded by a Marie-Curie-Grant and grants from the DAAD. CONTENTS 4.3.3 Approximation of functions having Baker domains . . .