Functions of genus zero for which the fast escaping set has Hausdorff dimension two

We study a family of transcendental entire functions of genus zero, for which all of the zeros lie within a closed sector strictly smaller than a half-plane. In general these functions lie outside the Eremenko-Lyubich class. We show that for functions in this family the fast escaping set has Hausdorff dimension equal to two

Simply connected fast escaping Fatou components

We give an example of a transcendental entire function with a simply connected fast escaping Fatou component, but with no multiply connected Fatou components. We also give a new criterion for points to be in the fast escaping set

Julia and escaping set spiders' webs of positive area

We study the dynamics of a collection of families of transcendental entire functions which generalises the well-known exponential and cosine families. We show that for functions in many of these families the Julia set, the escaping set and the fast escaping set are all spiders' webs of positive area. This result is unusual in that most of these functions lie outside the Eremenko-Lyubich class B. This is also the first result on the area of a spider's web

The MacLane class and the Eremenko-Lyubich class

In 1970 G. R. MacLane asked if it is possible for a locally univalent function in the class A to have an arc tract. This question remains open, but several results about it have been given. We significantly strengthen these results, in particular replacing the condition of local univalence by the more general condition that the set of critical values is bounded. Also, we adapt a recent powerful technique of C. J. Bishop in order to show that there is a function in the Eremenko-Lyubich class for the disc that is not in the class A

Entire functions for which the escaping set is a spider's web

We construct several new classes of transcendental entire functions, f , such that both the escaping set, I(f), and the fast escaping set, A(f), have a structure known as a spider’s web. We show that some of these classes have a degree of stability under changes in the function. We show that new examples of functions for which I(f) and A(f) are spiders’ webs can be constructed by composition, by differentiation, and by integration of existing examples. We use a property of spiders’ webs to give new results concerning functions with no unbounded Fatou components

Functions of genus zero for which the fast escaping set has Hausdorff dimension two

We study a family of transcendental entire functions of genus zero, for which all of the zeros lie within a closed sector strictly smaller than a half-plane. In general these functions lie outside the Eremenko-Lyubich class. We show that for functions in this family the fast escaping set has Hausdorff dimension equal to two