Fractal and multifractal structure of non-conformal repellers

Over the past 30 years research in the area of dimension and dynamics has developed enormously. This project will study the geometric structure and dimension of attractors and repellers of certain non-conformal dynamical systems, which often have a highly complicated fractal form.

A dynamical system essentially consists of a mapping on a set, such as a region of the plane, into itself. Of particular interest are the trajectories followed by repeatedly applying the mapping to some initial point. A repeller is a set that is mapped into itself, but such that nearby points outside the set move away from the set under iteration of the mapping. Repellers often have a fractal structure, with ever more intricate detail becoming apparent under continued enlargement. Dimension provides a way of measuring the size and complexity of such sets. Smooth geometrical shapes have dimensions that are whole numbers: curves are 1-dimensional, surfaces are 2-dimensional, etc., reflecting the number of coordinates needed to locate points in the object. To cope with the intricacy of fractal repellers, notions of dimension that need not be whole numbers are required, such as Hausdorff, packing and box-counting dimensions, and such fractional dimensions will play a key role in this project.

Much research has focused on conformal mappings, where the repellers are the often-pictured Julia sets of complex dynamics. These are generally locally self-similar, that is made up of many smaller, nearly similar copies of themselves, and their dimensions often may be found by solving an equation known as the Bowen-Ruelle formula.

Non-conformal mappings lead to locally self-affine repellers which may be of a very different character, with small parts of the repellers appearing increasingly elongated or distorted under enlargement. Given recent progress in the analysis of self-affine sets, which may be thought of as piecewise linear analogues of non-conformal repellers, it is timely to extend the dimension formulae to a non-conformal, nonlinear setting. However, finding dimensions of self-affine sets is far more awkward than for self-similar sets, not least because the dimensions need not vary continuously with the defining transformations. By analogy, it may be difficult to obtain dimension formulae that are valid in every situation, so a major part of the project will be to seek generic formulae that are 'almost always' valid in an appropriate sense. In the conformal setting, the thermodynamic formalism is an important tool for finding the dimensions, but for non-conformal systems a 'subadditive' version of the thermodynamic formalism will be needed.

Of course, it is desirable to know with certainty the dimension of specific repellers, rather than just finding formulae that are valid generically, so the project will also identify classes of functions for which the generic formulae can be guaranteed to give the actual value of dimension. A further line of investigation will consider distributions of mass across non-conformal repellers, with the aim of using analogues of fractional dimensions to quantify the irregularities of distribution, an area known as multifractal analysis.

The research proposed is theoretical in nature and, naturally, the immediate impact is likely to be mainly within the academic community. In the longer term research in this area, which impinges on many other parts of mathematics and science, could have consequences in unexpected directions.

The main interest will be amongst those working in dynamics, dimensions and fractal geometry, where research activity has vastly increased in recent years. Techniques developed to study both dimensions and dynamics are employed right across mathematics, for example in number theory (Diophantine analysis and beta-type transformations), differential equations (PDEs on fractal domains), stochastic processes (quantifying smoothness aspects and stochastic PDEs) and infinite group theory (automorphism groups of trees or Cantor sets), as well as in many areas of mathematical physics.

The area is widely applicable across science, including physical, biological, social and economic sciences, where dynamical systems arise in many contexts and where dimension is often used to quantify or detect phenomena. For example, in fluid dynamics dimensional estimates have been used to bound the degrees of freedom in models of turbulence, and in medicine methods of diagnosis based on estimating a fractal dimension have been proposed for a number of illnesses, including detecting various cancers and retinal disease. Whilst the proposed research is not directly aimed at such applications, its conclusions will be disseminated as widely as possible to reach researchers with diverse backgrounds. St Andrews University has a dedicated Knowledge Transfer Centre to assist in the development of any spin-offs that may be identified.

Thus the main pathway to achieving the maximum impact will be to ensure that the outcomes of the research quickly become as widely known and understood as possible. Preprints will be put on the arXiv and our institutional archive as soon as feasible, followed by publication in highly regarded mathematics journals. Ideas which may have a broader relevance may be published in journals such as 'Nonlinearity' which reach both the physics and mathematical communities or the fully interdisciplinary journal 'Fractals'. Ideas and results will also be presented at international conferences. These will include specialist meetings on dynamics and/or dimension that occur regularly, more general mathematics meetings and seminars, and also interdisciplinary meetings relating to fractals or dynamics that are organised from time to time to encourage cross-fertilisation between disciplines.

Passing on knowledge and scientific skills to future generations is crucial for our economic and social future, and St Andrews University places high importance on research-teaching linkages. Our School of Mathematics and Statistics offers final year undergraduate courses in Fractal Geometry, Dynamics and Ergodic Theory, and the nature of these subjects make it possible to intimate many state of the art ideas in the courses.

Some of the basic ideas of the area can be explained and illustrated at a very basic level - the underlying premise `performing a simple operation repeatedly can lead to extraordinarily intricate and beautiful structures' can easily be explained and attractively illustrated. I regularly give popular lectures at science festivals, schools, mathematical competition award ceremonies, etc., and this research will provide new attractive material that can be used to take mathematics into the wider community and enhance public understanding of the subject.

Currently the UK is internationally leading in the mathematics of dynamics and dimensions. This project will further enhance the UK's standing in the area and strengthen links with other research groups worldwide.