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Tiling spaces are Cantor set fiber bundles

Lorenzo Sadun·R. F. Williams
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Abstract

We prove that fairly general spaces of tilings of R^d are fiber bundles over the torus T^d, with totally disconnected fiber. This was conjectured (in a weaker form) in [W3], and proved in certain cases. In fact, we show that each such space is homeomorphic to the d-fold suspension of a Z^d subshift (or equivalently, a tiling space whose tiles are marked unit d-cubes). The only restrictions on our tiling spaces are that 1) the tiles are assumed to be polygons (polyhedra if d>2) that meet full-edge to full-edge (or full-face to full-face), 2) only a finite number of tile types are allowed, and 3) each tile type appears in only a finite number of orientations. The proof is constructive, and we illustrate it by constructing a `square' version of the Penrose tiling system.

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Lorenzo Sadun, R. F. Williams. 2018-07-09. Tiling spaces are Cantor set fiber bundles. https://arxiv.org/abs/math/0105125

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