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Details of Grant 

EPSRC Reference: EP/J022160/1
Title: Dimensions in complex dynamics: spiders' webs and speed of escape
Principal Investigator: Stallard, Professor G
Other Investigators:
Rippon, Professor PJ
Researcher Co-Investigators:
Project Partners:
Department: Mathematics & Statistics
Organisation: Open University
Scheme: Standard Research
Starts: 25 March 2013 Ends: 24 March 2015 Value (£): 178,755
EPSRC Research Topic Classifications:
Mathematical Analysis
EPSRC Industrial Sector Classifications:
No relevance to Underpinning Sectors
Related Grants:
Panel History:
Panel DatePanel NameOutcome
04 Jul 2012 Mathematics Prioritisation Panel Meeting July 2012 Announced
Summary on Grant Application Form
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.

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