Lyapunov Exponents and Spectral Properties of Aperiodic Structures

Order and disorder are familiar concepts to humans. Our brains have evolved to recognise and to appreciate aspects of symmetry and order in nature, as well as in architecture, arts or music. Essentially, science is about detecting and describing ordered patterns in the world around us, and understanding them in mathematical terms.

It is thus surprising that it appears to be difficult to give a precise mathematical definition of the concept of order, and indeed there is currently no generally accepted definition available. Consequently, we lack a complete understanding of what types of order are possible, and how to classify them.

It turns out to be useful to take a guide from nature. The type of order considered in this project is inspired by physics and crystallography, more precisely the surprising existence of intricately ordered materials called quasicrystals. Their discovery was acknowledged with the award of a Nobel Prize in Chemistry in 2011. As an abstraction of the order of atoms in such materials, mathematicians have investigated the order in patterns or tilings of space.

The project will consider a particular type of tilings, which are based on specific rules, in which a tiling can be constructed recursively. Some of these rules lead to particularly nice tilings, which are reasonably well understood. Here we mainly concentrate on tilings that lie outside this class, and investigate their properties, using a novel approach. This promises to provide new insight into order properties of such tilings, which would be a big step towards a classification of order in spatial structures, and will shed new light on the so-called Pisot substitution conjecture, one of the long-standing conjectures in the field that has so far eluded a proof.

The new approach also provides a link between two very different characterisations of aperiodic tilings by means of spectral properties. One of these is linked to diffraction as a measure of order, which uses a concept from crystallography and is intimately linked to a mathematical concept of spectrum used in dynamical systems theory. The other spectral characterisation is inspired by studying the physics of electron transport in aperiodic structures, and considers the electronic energy spectrum. These two spectral quantities behave rather differently, and the aim of the project is to understand this and relate these to each other.

While the proposed research is fundamental in nature, order phenomena are ubiquitous in nature and an improved understanding of order will be useful in many areas of science. Also, there are many potential applications of aperiodic structures of this type. The most promising are probably in manufactured materials, such as new light-weight strong materials with designed properties that could be used in engineering or medical applications. Aperiodic tilings often are also aesthetically appealing and have increasingly been used in arts and in architecture.

The proposed research is of a fundamental nature, so any application has to be seen as a long-term perspective. As with any fundamental research, it is of prime importance to inform the community working on potential applications about any developments and progress. This will be done by publishing in appropriate journals, and by attending the interdisciplinary conferences where experts meet. Publications will be made available to the wider public prior to publication via the arXiv, and subsequently by using open access journals and via the Open University research repository (ORO).

There are three main areas of (established or potential) impact of research in this field, to which the proposed project could contribute.

The first area of applications concerns technological applications of aperiodic systems. Purpose-made materials with designed properties have wide potential applications. These are not limited to naturally occurring structures, and with the possibility to manipulate systems at an atomic scale, there is a plethora of possible applications. Aperiodic structures have interesting properties and are likely to be superior to periodic and random structures in some situations (for instance due to their higher symmetry). The PI is currently involved in a project with colleagues in Design to investigate this in the context of additive manufacture of cellular structures. For aperiodic structures at the nanoscale, diffraction techniques will become relevant, and the results of this project may become important. Also, there are a number of potential applications around electronic spectral properties, such as for optical and electronic filters.

A second area of potential impact is in inspiring creative design. Some concrete examples are detailed in the Pathways to Impact section. This may be less likely to arise from this particular project, since the emphasis is on spectral properties not new structures, but it is conceivable that some particularly appealing structures may arise in connection with the final theme of this project, which could be of interest. The PI hopes to establish further contacts to artists through including arts content in the forthcoming ICMS workshop.

Public engagement with mathematics is important, in particular with regard to enthusing young people to take up mathematics at school and higher education level. There is a considerable shortage of mathematicians in the UK, particularly (but not only) in education, and the only way to change this is a long-term strategy to increase the take-up of the subject. Aperiodic order is an ideal field for engagement activities, due to its aesthetic appeal and the possibility to explain complex mathematical theory in terms of pictures in a way that makes the ideas accessible to a lay audience. This impact is very likely to be achieved, as the PI has a strong track record in public engagement, and will continue to take an active approach towards engagement. The Open University and the School of Mathematics and Statistics are very supportive of outreach activities, and the School is currently further developing its proactive approach in this area. Concerning the proposed research, the aspect that is most likely to be amenable for public engagement is that of characterising different types of (aperiodic) order, and distinguishing them by properties that are linked to applications.