| EPSRC Reference: |
EP/L019841/1 |
| Title: |
Iteration of quasiregular mappings |
| Principal Investigator: |
Nicks, Dr DA |
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| Department: |
Sch of Mathematical Sciences |
| Organisation: |
University of Nottingham |
| Scheme: |
First Grant - Revised 2009 |
| Starts: |
15 September 2014 |
Ends: |
14 September 2016 |
Value (£): |
98,917
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| No relevance to Underpinning Sectors |
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Summary on Grant Application Form |
Complex dynamics is the study of iteration of analytic and meromorphic functions on the complex plane. Inspired in part by the fascinating computer-generated images of the intricate fractal sets involved, the last thirty years have seen a great resurgence of interest in complex dynamics amongst mathematicians and the wider public. Central to this theory is the Julia set, consisting of those points in the plane at which the iterates behave chaotically.
Quasiregular mappings provide a natural generalisation of analytic functions to n-dimensional Euclidean space. At an infinitesimal scale, analytic functions always map circles to circles, whereas quasiregular functions are permitted to send infinitesimal spheres to infinitesimal ellipsoids of bounded eccentricity. This means that quasiregular maps are considerably more flexible than analytic functions, even in two dimensions. Despite this greater generality, many theorems of classical complex analysis have counterparts in the quasiregular setting. While a well-established literature exists on the function theory of quasiregular mappings, much less is known about their iterative behaviour. This project will explore the extent to which the results and successes of complex dynamics can be transferred to the emerging theory of quasiregular iteration, and aims to uncover the similarities and differences between these two related fields.
One major difficulty within quasiregular dynamics is that the amount of local stretching can grow as the number of iterates increases. Problems such as this demonstrate that there are fundamental differences to the analytic case. Nonetheless, recent research has shown, perhaps surprisingly, that some features of complex dynamics do persist in the quasiregular setting. For example, it is possible to define an analogue of the Julia set for quasiregular maps. The proposed research will investigate the structure and properties of this new Julia set as well as studying the 'escaping set' of points whose images tend to infinity under iteration. The escaping set, and variations upon it, are currently playing an increasingly important role in the study of iteration of transcendental entire functions.
This project will help to establish a framework for quasiregular dynamics, which promises to be an exciting and fertile area of research due to the variety of intriguing questions that arise via the analogy to complex dynamics.
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Key Findings |
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This information can now be found on Gateway to Research (GtR) http://gtr.rcuk.ac.uk
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Potential use in non-academic contexts |
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This information can now be found on Gateway to Research (GtR) http://gtr.rcuk.ac.uk
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Impacts |
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| Description |
This information can now be found on Gateway to Research (GtR) http://gtr.rcuk.ac.uk |
| Summary |
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| Date Materialised |
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Sectors submitted by the Researcher |
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This information can now be found on Gateway to Research (GtR) http://gtr.rcuk.ac.uk
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| Project URL: |
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| Further Information: |
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| Organisation Website: |
http://www.nottingham.ac.uk |