Applications of space filling curves to substitution tilings

二十世纪末最壮观的科学发现之一是一种既不是晶态也不是非晶态的新材料。这导致了结晶学的范式转变，这些合金现在被称为准晶。准晶是由被称为非周期平铺的图案在数学上建模的，这种图案缺乏通常意义上的对称性，但仍然显示出长程有序。最著名的例子是罗杰·彭罗斯爵士，他的瓷砖显示出与丹·谢克特曼教授发现的准晶相同的“不可能”的对称性。这一方案的研究开创了一种通过降维来研究非周期平铺的新方法。数字革命通过将二维甚至三维图像编码为0和1的序列，使其在从一个地方发送到另一个地方方面取得了深刻的进步。这项建议中的研究与此类似，只是图像是无限的，不需要以局部系统的方式排列像素。特别是，这项提案中的研究通过使用空间归档曲线启动了对非周期瓷砖的类似类型的编码；这种类型的分形是在18世纪末发现的，有助于重塑我们对大小、面积和体积的数学概念。与视频源的方式大致相同，一些信息通过编码进行压缩。然而，我们仍然可以获得关于原始平铺的大量信息，特别是当空间填充曲线来自最初用于定义平铺的基本方法时。值得注意的是，在非周期瓷砖的情况下，所有关于瓷砖如何拼接在一起的几何信息都被编码在数字序列中，这使得从数学角度来看它非常容易处理；它是一个纯粹的组合对象。对于每个非周期瓷砖，我们定义了一个动力系统，该系统由拓扑空间上的映射组成，该拓扑空间上的各个点是无限瓷砖。证明了这个拓扑空间是一个环面上的Cantor集纤维丛，也就是说，它是一个具有任意多个孔的环面，它的表面上的每一点都有分形。这个空间的奇特性质使它的研究变得极其困难。因此，拓扑不变量和算子代数不变量一直是平铺空间研究的重点。该方案通过研究与空间填充曲线相关的组合空间，为研究这种动力系统提供了一种新的思路，这种方法简单得多，同时保留了更复杂系统的大部分信息。该方案所采用的新方法将对非周期平铺理论的研究产生影响，甚至对双曲动力系统、算子代数和分形几何的更广泛的研究产生影响。

One of the most spectacular scientific discoveries of the late twentieth century was a new material that was neither crystalline nor amorphous. This created a paradigm shift in crystallography, and these alloys are now called quasicrystals. Quasicrystals are modelled mathematically by patterns called aperiodic tilings that lack symmetry in the usual sense, but still exhibit long-range order. The most famous example is due to Sir Roger Penrose, whose tiling exhibited the same `impossible' symmetry as the quasicrystals discovered by Professor Dan Shechtman. The research in this proposal initiates a new method of studying aperiodic tilings through dimension reduction.The digital revolution has made profound advances in sending two- and even three-dimensional images from place to place by encoding them as a sequence of zeros and ones. The research in this proposal draws analogy with this except that the image is infinite and need not have pixels arranged in a locally systematic way. In particular, the research in this proposal initiates a similar type of encoding of an aperiodic tiling through the use of space-filing curves; a type of fractal that was discovered in the late 18th century that helped to reshape our mathematical notions of size, area and volume. In much the same way as a video feed, some information is compressed through the encoding. However, we can still garner a vast amount of information about the original tiling, especially when the space filling curve comes from the underlying method used to define the tiling in the first place. Significantly, in the case of aperiodic tilings, all the geometric information about how tiles fit together is encoded in the digital sequence making it very easy to work with from a mathematical perspective; it is a purely combinatorial object.To each aperiodic tiling we define a dynamical system that consists of a map on a topological space whose individual points are infinite tilings. It has been shown that this topological space is a Cantor set fibre bundle over a torus; that is, it is a donut with an arbitrary number of holes that has fractals emanating from every point on its surface. The bizarre nature of this space makes it extremely difficult to study. For this reason, topological and operator algebraic invariants have been the focus of research on tiling spaces. The programme of research outlined in this proposal gives a new attack on studying this dynamical system by studying the combinatorial space associated with the space filling curve, which is much simpler while retaining most information about the more complicated system.The new approach taken in this proposal will have impact across research in aperiodic tiling theory, and even to the more general study of hyperbolic dynamical systems, operator algebras and fractal geometry.