Celebratio Mathematica — Williams

An Autobiography

Works connected to Lorenzo Adlai Sadun

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L. Sadun and R. F. Williams: “Tiling spaces are Cantor set fiber bundles,” Ergodic Theory Dynam. Systems 23 : 1 (February 2003), pp. 307–316. MR 1971208 Zbl 1038.37014 article

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

the tiles are assumed to be polygons (polyhedra if \( d > 2 \)) that meet full-edge to full-edge (or full-face to full-face),
only a finite number of tile types are allowed, and
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.