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Applications of space filling curves to substitution tilings

空间填充曲线在替代平铺中的应用

基本信息

批准号：
EP/R013691/1
负责人：
Michael Whittaker
金额：
$ 12.88万
依托单位：
University of Glasgow
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2018
资助国家：
英国
起止时间：
2018 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FR013691%2F1
关键词：
Applications space filling curves substitution

项目摘要

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.

20世纪末最引人注目的科学发现之一是一种既不是晶体也不是无定形的新材料。这创造了晶体学的范式转变，这些合金现在被称为准晶体。准晶在数学上被称为非周期性镶嵌的图案所模拟，这种图案缺乏通常意义上的对称性，但仍然表现出长程有序。最著名的例子是罗杰·彭罗斯爵士（Sir Roger Penrose），他的瓷砖展示了与丹·谢赫特曼（Dan Shechtman）教授发现的准晶体相同的“不可能”对称性。该提案的研究开创了一种通过降维来研究非周期性镶嵌的新方法。数字革命在通过将二维甚至三维图像编码为0和1的序列来将其从一个地方发送到另一个地方方面取得了深刻的进展。本提案中的研究与此类比，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填充曲线启动了一种类似的非周期性瓷砖编码;这是一种在18世纪后期发现的分形，有助于重塑我们对大小，面积和体积的数学概念。与视频馈送的方式大致相同，一些信息通过编码进行压缩。然而，我们仍然可以获得大量关于原始平铺的信息，特别是当空间填充曲线来自于最初用于定义平铺的底层方法时。值得注意的是，在非周期性平铺的情况下，所有关于平铺如何组合在一起的几何信息都被编码在数字序列中，这使得它很容易从数学的角度进行工作;它是一个纯粹的组合对象。对于每个非周期性平铺，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，其各个点是无限平铺。已经证明，这个拓扑空间是环面上的康托集纤维丛;也就是说，它是一个具有任意数量孔的圆环，其表面上的每一点都有分形。这个空间的奇异性质使它非常难以研究。因此，拓扑不变量和算子代数不变量一直是瓦片空间研究的重点。该研究计划通过研究与空间填充曲线相关的组合空间，为研究该动力系统提供了一种新的方法，这种方法简单得多，同时保留了更复杂系统的大部分信息。该研究计划中采用的新方法将对非周期镶嵌理论的研究产生影响，甚至对双曲动力系统的更一般的研究产生影响，算子代数和分形几何。