Applications of space filling curves to substitution tilings

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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.