Complexity of random substitution tilings

准晶体的发现——荣获2011年诺贝尔化学奖——令物理学家、化学家和材料科学家感到意外。准晶体是一种具有长程有序但缺乏平移对称性的结构，从催化剂到低摩擦涂层，再到量子计算中的理论应用，准晶体已经有了丰富的应用。与许多结构一样，大多数物理准晶体并不完美，并表现出局部缺陷。因此，好的准晶体理论模型应该同时表现出远程有序和局部无序的特征。这种模型是由随机替换的平铺空间提供的。在这里，我们将开始开发这些模型的急需和实质性的层次分类。拓扑熵是动态系统的不变量，它量化了系统的复杂性。在整个项目中，我们将利用一维符号随机替换系统的最新发展，结合确定性替换平铺空间的丰富理论，为随机替换平铺空间开发一个鲁棒的拓扑熵理论。此外，我们将考虑一种新的方法来计算二维或更高维度的平铺空间的复杂性，通过量化由原始系统沿超平面切片引起的低维子系统的复杂性。这将提供一种识别具有相同拓扑熵的贴图的方法。因此，创建了一种新的方法来区分和排序随机替换平铺空间的复杂性，因此，采取了建立分层分类的初步步骤。这个项目的主题可以在非周期秩序之外的许多数学领域中找到，包括动力系统，分形几何，几何测量理论，调和分析和数论。此外，由于随机替代平铺空间为准晶体和分形渗透等物理现象提供了理论模型，我们期望我们开发的技术将对数学以外的领域产生持久的影响。

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.