Complexity of random substitution tilings

准晶体的发现--被授予2011年诺贝尔化学奖--让物理学家、化学家和材料科学家感到惊讶。准晶是具有长程有序但缺乏平移对称性的结构，并且已经发现了大量的应用，从催化剂到低摩擦涂层，再到量子计算中的理论应用。与许多结构一样，大多数物理准晶并不完美，并且表现出局部缺陷。因此，好的准晶理论模型应该同时表现出长程有序和局域无序的特征。这种模型是由随机替换的平铺空间提供的。在这里，我们将开始发展一个急需的和实质性的层次分类这些models.Topological熵是一个不变量的动力系统，量化其复杂性。在整个项目中，我们将开发一个强大的随机替代瓦片空间的拓扑熵理论，利用最近的发展，在一维符号随机替代系统，结合确定性替代瓦片空间的丰富理论。此外，我们将考虑一种新的方法，这种平铺空间的复杂性在二维和更高的，通过量化的复杂性低维子系统引起的切片沿沿着一个超平面的原始系统。这将提供一种识别具有相同拓扑熵的镶嵌的方法。因此，创建一个新的方法来区分和排序随机替换平铺空间的基础上，他们的复杂性，并因此，采取初步步骤建立一个层次的classification.The主题的这个项目可以在许多数学领域以外的非周期秩序，包括动力系统，分形几何，几何测度理论，调和分析和数论。此外，由于随机置换平铺空间为准晶和分形渗流等物理现象提供了理论模型，我们预计我们开发的技术将产生超越数学的持久影响。

The discovery of quasicrystals - honoured with the 2011 Nobel Prize in Chemistry - came as a surprise to physicists, chemists and materials scientists. Quasicrystals are structures which possess long-range order but lack translational symmetry and have found a wealth of applications, from catalysts to low friction coatings, over to theoretical applications in quantum computing. As with many structures, most physical quasicrystals are not perfect, and exhibit local defects. Thus, good theoretical models for quasicrystals should exhibit features of both long-range order and local disorder. Such models are provided by tiling spaces of random substitutions. Here, we will initiate the development of a much-needed and substantial hierarchical classification of these models.Topological entropy is an invariant of dynamical systems which quantifies their complexity. Throughout this project, we will develop a robust theory of topological entropy for random substitution tiling spaces, leveraging recent developments for symbolic random substitution systems in one-dimension, combined with the rich theory of deterministic substitution tiling spaces. Moreover, we will consider a novel approach to complexity of such tiling spaces in dimensions two and higher, by quantifying the complexity of lower-dimensional subsystems induced by slicing the original system along a hyperplane. This will provide a method of discerning tilings which have the same topological entropy. Thus, creating a new means of distinguishing and ordering random substitution tiling spaces based on their complexity, and hence, taking the initial steps towards establishing a hierarchical classification.The themes of this project can be found in many mathematical fields outside of aperiodic order, including dynamical systems, fractal geometry, geometric measure theory, harmonic analysis and number theory. Further, as random substitution tiling spaces provide theoretical models for physical phenomena, such as quasicrystals and fractal percolation, we expect that the techniques we develop will have a lasting impact beyond mathematics.