Dimensions in complex dynamics: spiders' webs and speed of escape

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 and the Julia set.

Much work in this area has focused on obtaining an understanding of the sizes of these sets and significant subsets as measured by their Hausdorff dimensions. This has led to fundamental insights into the nature of these sets with some results being completely unexpected. Most work to date has, however, focused on functions in the so called class B for which a range of powerful techniques are avaliable.

Recently, however, the first estimates for dimensions of these sets for functions outside of the class B have been obtained. Moreover, the investigators have introduced new techniques to the area and shown that there are many functions outside the class B for which the escaping set has a novel 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 obtain a greater understanding of this structure.

It is therefore timely to begin a programme of research investigating the size of the escaping and Julia sets for functions outside the class B. This project will focus on those functions for which the escaping set has the structure of a spider's web and also on two significant subsets of the escaping set, namely the fast escaping set which has been shown to play a key role and the slow escaping set which has only recently been introduced into the subject. The proposed research aims to build upon the new techniques recently introduced to the area in order to establish a framework which will form the foundation for future research in this area.

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.

In order to maximise the chances of others using the techniques developed during the project, the results of the research will be disseminated as widely as possible. They will be published in high quality mathematical journals and will also be made available on the arXiv and on The Open University's open access institutional repository - Open Research Online (ORO) at http://oro.open.ac.uk. ORO is now one of the largest repositories in the UK. The site receives an average of 40,000 visitors per month from over 200 different countries and has received over 1.6 million visitors since 2006. It enables access to research outputs via common search engines including Google, by using the OAI (Open Archives Initiative) Protocol for Metadata Harvesting.

The results of the research will be presented at international conferences and at international workshops for early career researchers in the area. This is an active area of research and a number of such events are liekely to occur during the lifetime of this project. The investigators have played a leading role at many of these events in recent years; as organisers, invited speakers and presenters of courses at workshops. They have recently submitted a proposal to organise a conference on the relationship between complex analysis and complex dynamics to be held at the ICMS in Edinburgh in 2013.

The investigators and Professor Bergweiler 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 may well 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.
Dr Peter has expertise in this area and will produce further pictures of this type as part of the research project thus developing skills which will be of use to him in a range of situations. The investigators will ensure that these pictures are used 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 and strengthen links with other research groups in Europe.