Iteration of quasiregular mappings

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.

The proposed research project lies within the area of pure mathematics, a subject whose long-term effect on many fields of science and engineering is indisputable, although history shows that these impacts often occur in surprising ways that were unlikely to be predicted at the time the fundamental underlying research was carried out. This individual project will contribute to the strength and vitality of pure mathematics as a discipline and it is this, taken as a whole, that will have powerful, enduring economic and societal impacts far into the future.

The investigation lies at the junction between complex dynamics and quasiregular function theory, and so the short term impact within the academic community is likely to be greatest amongst researchers in these two fields. As described in the case, the project will help to develop the newly-emerging topic of general quasiregular dynamics that has the potential to be a fertile area in which others can obtain further results in future.

The use of computer modelling and numerical approximation techniques to study real-world systems is now widespread and it is often iterative processes that underpin such methods. The proposed research aims to extend our understanding of iteration in one complex variable to a broader and more flexible setting, where the iteration takes place in several real variables. This is closer to the situation encountered in modelling and approximation applications and thus it is conceivable that work connected to this project could have some future influence in this direction.

There will be a significant positive impact on the career of the postdoctoral research assistant appointed to work on this project. In addition to specific mathematical training, they will gain valuable research skills and experience that will bring benefits to future employers and collaborators whether within or outside academia. The postdoc's career prospects, and indeed those of the PI himself, will be enhanced by their contributions to the published outputs of this project, and the opportunity to attend national and international conferences will allow them to raise their own personal profiles and to promote their research to a wide audience.