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Tiling spaces are Cantor set fiber bundles
We prove that fairly general spaces of tilings of \mathbb{R}^d are fiber bundles over the torus T^d, with totally disconnected fiber. This was conjectured (in a weaker form) in the second author's recent work, and proved in certain cases. In fact, we show that each such space is homeomorphic to...
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| Published in: | Ergodic theory and dynamical systems 2003-02, Vol.23 (1), p.307-316 |
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| Main Authors: | , |
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| Language: | English |
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| container_end_page | 316 |
| container_issue | 1 |
| container_start_page | 307 |
| container_title | Ergodic theory and dynamical systems |
| container_volume | 23 |
| creator | SADUN, LORENZO WILLIAMS, R. F. |
| description | We prove that fairly general spaces of tilings of \mathbb{R}^d are fiber bundles over the torus T^d, with totally disconnected fiber. This was conjectured (in a weaker form) in the second author's recent work, and proved in certain cases. In fact, we show that each such space is homeomorphic to the d-fold suspension of a \mathbb{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. |
| doi_str_mv | 10.1017/S0143385702000949 |
| format | article |
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| source | Cambridge Journals Online |
| title | Tiling spaces are Cantor set fiber bundles |
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