Topics in Transcendental Dynamics

Sixsmith, David (2013). Topics in Transcendental Dynamics. PhD thesis The Open University.



We study the iteration of a transcendental entire function, f; in particular, the fast escaping set, A(f). This set consists of points that iterate to infinity as fast as possible, and plays a significant role in transcendental dynamics.

First we investigate functions for which A(f) has a structure called a spider's web. We construct several new classes of function with this property. We show that some of these classes have a degree of stability under changes in the function, and that new examples of functions with this property 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.

When A(f) is a spider's web, it contains a sequence of fundamental loops. We next explore the structure of these fundamental loops for functions with a multiply connected Fatou component, and show that there exist functions for which some fundamental loops are analytic curves and approximately circles, while others are geometrically highly distorted. We do this by introducing a real-valued function which measures the rate of escape of points in A(f), and show that this function has a number of interesting properties.

Next we study functions with a simply connected Fatou component in A(f). We give an example of a function with this property, which - in contrast to the only other known functions of this type - has no multiply connected Fatou components. To do this we also prove a new criterion for points to be in A(f).

Finally, we investigate the much studied Eremenko-Lyubich class of transcendental entire functions with a bounded set of singular values. We give a new characterisation of this class, and a new result regarding direct singularities which are not logarithmic.

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