Project description
Complex dynamics is the study of iteration of analytic functions on the complex plane. A rich mathematical structure is seen to emerge amidst the chaotic behaviour. Its appeal is enhanced by the intricate nature of the Julia sets that arise, and fascinating images of these fractal sets are widely admired.
Quasiregular mappings of n-dimensional real space generalise analytic functions on the complex plane. Roughly, a mapping is called quasiregular if it locally distorts space by only a bounded amount, so that small spheres are mapped to small ellipsoids. This is more flexible than the situation with analytic functions, where the Cauchy-Riemann equations tell us that infinitesimally small circles are mapped to small circles.
There are many similarities between the behaviour of analytic functions and quasiregular mappings. One can therefore attempt to develop a theory of quasiregular iteration parallel to the results of complex dynamics. Such a theory is just beginning to emerge, lying between the well-studied analytic case (where many powerful tools from complex analysis are available) and general iteration in several real variables, which is much less well-understood.
The problems studied will be inspired and guided by existing results in complex dynamics. For example, we can ask questions about the ‘escaping set’ of a function – this is the set of all starting points from which the sequence of iterates tends to infinity. One of the challenges we encounter is that as we increase the number of iterations of a quasiregular mapping, the amount of local distortion may become increasingly large.
This is very much a pure mathematics project and will appeal to someone who enjoys topics such as real analysis, complex analysis, metric spaces or discrete dynamical systems.