Baker's conjecture and Eremenko's conjecture: a unified approach.

The proposed research is in the area of complex dynamics which has experienced explosive growth in the last 25 years following the advent of computer graphics. For each meromorphic function, the complex plane is split into two fundamentally different parts - the Fatou set, where the behaviour of the iterates of the function is stable under local variation, and the Julia set, where it is chaotic. Computer pictures demonstrate that most Julia sets are highly intricate.Another key object of study is the escaping set which consists of the points that escape to infinity under iteration. This set plays a major role in complex dynamics since the Julia set is equal to the boundary of the escaping set. For polynomials, the dynamics on the escaping set are relatively simple, but for transcendental entire functions the escaping set is much more complex. In order to make progress in the area of transcendental complex dynamics it is essential to gain a greater understanding of the structure of the escaping set.One of the key questions in this area is whether all the components of the escaping set are unbounded - this is now known as Eremenko's conjecture and has attracted a great deal of interest. Much work on this question has centred on identifying functions for which the escaping set has a structure known as a Cantor bouquet of curves. The investigators have introduced new techniques to the area and shown that there are many functions for which the escaping set has a very different structure described as an infinite spider's web. They have further shown that the existence of certain types of spiders' webs implies several strong properties and so it is highly desirable to identify as many functions as possible for which the escaping set has this structure.Another question in transcendental dynamics that has attracted much interest is whether functions of small growth have no unbounded components of the Fatou set - this is now known as Baker's conjecture. The investigators have discovered a surprising connection between these two conjectures and shown that the techniques used to make progress on Baker's conjecture are precisely what is needed to show that the escaping set is a spider's web of the type mentioned above. Further, if the escaping set has this form then both Eremenko's conjecture and Baker's conjecture hold.The object of the proposed research is to to build upon these new techniques and ideas to make substantial progress on both conjectures. The work will lead to an increased understanding of the structure of the escaping set for large classes of functions, and this will enable progress to be made on many other questions in transcendental dynamics.