Lyapunov Exponents and Spectral Properties of Aperiodic Structures

秩序和无序是人类熟悉的概念。我们的大脑已经进化到能够识别和欣赏自然界、建筑、艺术或音乐中的对称性和秩序的各个方面。从本质上讲，科学是关于检测和描述我们周围世界中的有序模式，并用数学术语理解它们。因此，令人惊讶的是，似乎很难给出秩序概念的精确数学定义，事实上，目前还没有普遍接受的定义可用。因此，我们对哪些类型的秩序是可能的，以及如何对它们进行分类缺乏完整的理解。事实证明，从自然中寻找向导是很有用的。这个项目中考虑的有序类型的灵感来自物理学和结晶学，更准确地说，令人惊讶的是存在着被称为准晶的复杂有序材料。他们的发现在2011年获得了诺贝尔化学奖。作为这种材料中原子顺序的抽象，数学家们研究了空间图案或瓷砖的顺序。该项目将考虑一种特定类型的瓷砖，它基于特定的规则，其中瓷砖可以递归构造。其中一些规则导致了特别好的平铺，这是相当好的理解。在这里，我们主要集中在这个类之外的平铺，并使用一种新的方法来研究它们的性质。这有望为此类平铺的有序性质提供新的见解，这将是向空间结构中的有序分类迈出的一大步，并将为所谓的皮索替代猜想提供新的线索，这是该领域长期存在的猜想之一，但迄今尚无证据。新的方法还通过光谱特性在非周期平铺的两种截然不同的特征之间提供了联系。其中之一与作为有序度量的衍射有关，它使用了结晶学中的一个概念，并与动力系统理论中使用的光谱的数学概念密切相关。另一种光谱特征是从研究非周期结构中的电子传输物理中得到启发的，并考虑了电子能谱。这两个光谱量的行为相当不同，该项目的目的是理解这一点并将它们相互联系。虽然拟议的研究在本质上是基本的，但秩序现象在自然界中无处不在，对秩序的更好理解将在许多科学领域中有用。此外，这种类型的非周期结构还有许多潜在的应用。最有希望的可能是制造材料，例如具有可用于工程或医疗应用的设计性能的新型轻质坚固材料。非周期性瓷砖通常在美学上也很有吸引力，越来越多地被用于艺术和建筑中。

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.