Tiling spaces are Cantor set fiber bundles

Abstract

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.