Baker's conjecture and Eremenko's conjecture: new directions

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 complicated. In order to make progress in the area of transcendental complex dynamics it is essential to gain a greater understanding of the structure of these fundamental sets.

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. 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 discovered a surprising connection between these two conjectures and showed that, for a large class of functions, both Baker's conjecture and Eremenko's conjecture hold, with the escaping set having a novel structure described as an infinite spider's web.

This connection was discovered by considering the so called `fast escaping set' of points that escape to infinity faster than the iterated maximum modulus. This set is now known to play a key role in transcendental dynamics and all previous work on Baker's conjecture has focused on points in this set. The investigators have recently shown, however, that, in order to solve Baker's conjecture, it is necessary to consider points that escape to infinity more slowly.

One of the aims of this project is to consider the set of points that escape to infinity faster than the iterated minimum modulus. The proposal to consider this set is highly novel and has the potential to transform our understanding of the structure of the escaping set.

When considering points that escape to infinity at slower rates, it is necessary to introduce completely new techniques in order to demonstrate that the escaping set has the structure of a spider's web. The investigators have recently shown that, in some situations, this can be achieved by using a variety of techniques from complex analysis to prove that the images of certain curves wind many times round the origin.

The object of the proposed research is to build upon these new techniques and ideas to make substantial progress on both conjectures. Moreover, it may be possible to show that one of the conclusions of Baker's conjecture holds much more generally than was envisaged when the conjecture was made. The work will lead to new results of general interest in both complex dynamics and complex analysis and to new interactions between the two areas.

The proposed research is in the area of pure mathematics and so, like most research in this area, it is likely that the main impact in the short term will be within the academic community. It is hard to judge what the longer term impacts might be but, in order to maximise the chances of impact to other areas, the results of the research will be publicised as widely as possible.

Within the academic community, the initial impact will be greatest amongst researchers working in complex dynamics. This work is focused on two new significant developments (techniques to give a new winding property and the consideration of the iterated minimum modulus) and will provide the foundation for a new body of work in these two areas. It will lead to an increased understanding of these concepts and other new sets such as the spider's web escaping set and of the role which these can play in transcendental dynamics.

The investigators also have a record of proving general results in complex analysis in order to make progress on problems in complex dynamics. These complex analysis results can have much wider applications than to the problem in complex dynamics which provided the original motivation. For example, they have proved results in this way which have then been used by others to make progress on problems involving differential equations. The research for this project is expected to lead to more results of this type with wider applications.

Work in this area leads to highly intricate computer pictures which attract great interest. A picture of the first spider's web discovered by the investigators appeared in an exhibition at the 2008 British Association Festival of Science in Liverpool and in an audio slideshow inspired by this event that can be seen on the BBC website. These pictures help to make the results of the research accessible to as wide an audience as possible and to inspire others to carry out mathematical research.

The UK is currently world leading in the area of transcendental dynamics - this project would further establish the UK's reputation in this area. It would also strengthen the interactions between the complex dynamics and complex analysis communities.