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Details of Grant 

EPSRC Reference: EP/R010560/1
Title: Classifying Wandering Domains
Principal Investigator: Stallard, Professor G
Other Investigators:
Rippon, Professor PJ
Researcher Co-Investigators:
Project Partners:
Department: Faculty of Sci, Tech, Eng & Maths (STEM)
Organisation: Open University
Scheme: Standard Research
Starts: 01 April 2018 Ends: 31 March 2021 Value (£): 390,936
EPSRC Research Topic Classifications:
Mathematical Analysis
EPSRC Industrial Sector Classifications:
No relevance to Underpinning Sectors
Related Grants:
Panel History:
Panel DatePanel NameOutcome
06 Sep 2017 EPSRC Mathematical Sciences Prioritisation Panel September 2017 Announced
Summary on Grant Application Form
The proposed research is in the field of complex dynamics which has experienced explosive growth in the last 30 years, with the advent of computer graphics demonstrating the highly intricate nature of the sets involved, and with the introduction of many deep techniques from complex analysis.

This research project will seek to achieve the ambitious objective of completing the classification of the different types of dynamical behaviour that can occur within the components of the Fatou set. (The Fatou set of an analytic function is the set where the behaviour of the iterates of the function is stable.) A complete classification of Fatou components exists for rational functions and this is fundamental to work on rational dynamics. Until recently, such a classification for transcendental entire functions seemed out of reach.

The background to the project is that a complete classification of periodic Fatou components for rational functions was given by the founders of complex dynamics, Fatou and Julia, nearly 100 years ago. For many years it was unknown whether there are other Fatou components which never map into a periodic component - such components are known as wandering domains.

One of the most famous results in complex dynamics is Sullivan's `no wandering domains theorem' published in the Annals in 1982, which shows that, for rational functions, all Fatou components are eventually periodic. A major difference between rational dynamics and transcendental dynamics is that wandering domains can exist for transcendental functions. There is currently no general description of the dynamical behaviour inside wandering domains.

The first example of a wandering domain was given by Baker in 1976. His example was multiply connected and he later showed that many of the basic geometric properties of this example hold for all multiply connected wandering domains.

In a recent major paper, the investigators together with Walter Bergweiler gave a remarkably complete description of the dynamical behaviour in multiply connected wandering domains. This holds out the prospect, for the first time, that it might be possible to give a complete description of the dynamical behaviour in simply connected wandering domains. The purpose of this project is to bring this prospect to fruition, thus giving a complete classification of Fatou components.

There are many types of simply connected wandering domains, and so the task of giving a complete description of the dynamical behaviour in such domains will be much more challenging than for multiply connected wandering domains. We will begin by analysing examples of wandering domains that can be thought of as escaping versions of the different types of periodic Fatou components, and using a range of techniques to construct new examples. By studying the limiting behaviour of the hyperbolic distance between pairs of points in such domains we hope to produce a classification of the different types of behaviour that are possible. This should enable us to identify sequences of absorbing domains inside the wandering domains, within which the dynamics behave in a specified way. We will then address the more challenging task of constructing new examples of wandering domains that do not escape and classifying such domains.

Our classification should provide new insight into major open problems in complex dynamics which we will explore in the latter part of the project. In particular, it could lead to a resolution of the question as to whether commuting analytic functions have equal Julia sets. This would be a key step towards addressing the fundamental question as to which pairs of analytic functions commute.

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